All right, so now we've got our ramp in place. Let's go and start focusing on the deck portion of the the quarter pipe Okay, let's focus on this deck area So it's kind of like this this area where like the skateboarders will stand or the inline skaters will stand at the top here What waiting for their turn? All right And so what we need for this is I need that top curve again because we're gonna start from there All right, so we're gonna get this top curve up here and that comes from Where do we get that top curve? Oh, sorry, it's over here. So we want to get it from here. All right, so let's do that Yeah, so what I'm gonna do is drop down a null node right here and we're gonna say out Top curve. All right. So notice how we're starting to utilize work that we've already done. I mean, that's The more and more you start using the nodes and Houdini you'll start doing this So you're not having to redo all this work. I mean the lines are to there. So we might as well just start from there Okay, so let's make a object merge node and start a new little thread here or new little stream All right, so I'm gonna say get Top curve like so and I'm gonna click and drag this guy into that Object one slot and then turn off that import transform option All right. So now we've got the curve there so we can work just on this guy and stay focused All right, so let's drop down a transform node and what I'm gonna do is this is going to basically determine the Deck depth. All right, so this will be our deck depth option here And what we're gonna do is we're gonna control that with this z value. So I'm gonna push it back in Let's say negative point five so half a meter. All right, so now we've got these two curves here That allows us to create the deck geometry. All right Cool. So one thing that we need to do now and I should actually get rid of these points Let's do that. Let me show you guys how to do that So I'm gonna drop down a facet node and I don't really need this extra We could always re-samb out later on if we if we really need to so what I'm gonna do is inside of this facet node I'm gonna remove inline points and that works because those points in the middle there are literally in line with each other All right, it won't work if it's actually curved. They need to be in line But it works in this case. So nice quick little fix for that. So now we've got a clean curve All right, so let's merge these two guys together. So I'm just gonna select them both I'm gonna hold down alt on the keyboard I'm gonna hover my mouse over one of these outputs and then left click and drag and that automatically creates a merge node for us So now I've merged both those curves together Okay, and what I'm gonna do is create an ad node because I really just need kind of the geometry on the outside here I don't need the geometry on the inside here. At least not for the way that I'm modeling this particular ramp So with this ad node, I can actually go and delete all the geometry So it leaves me with just the point. So it's this option right here. I'm gonna select that polygons tab I'm gonna go to buy group and I'm gonna set this to skip every nth point And what that does is it gives me the side curves. All right So now I can actually take this deck depth curve right here and these two side curves and I can create one path All right, so let's do that. So I'm gonna select this guy and I'm gonna select this guy Again, I'm gonna hover over the output of this ad node and then left click and drag to create another merge node And I'm gonna actually sort my Inputs here so that it's a little bit cleaner to do that select the merge node and hit this little up arrow there So look at that. We now have a you know full curve But if I turn on my point numbers, we actually have overlapping points here So we need to take care of that. I need to actually fuse those guys back together Like so. All right, so there we go. So now I've got a nice curve But if you actually look at even further if I turn on my primdoms I actually have three primitives and I actually need just a single curve or single primitive So at this point what we can do is we can get a join node So I'm going to take a join node here And then I'm going to basically set the only connected option And look at that. We now have a single primitive and fuse points. So 0 1 2 3 awesome Cool. So with that I can build now the the back of the geometry So I can actually go and create another transform node here And inside of this transform node I'm just going to drop the geometry all the way to the floor and to do that really quickly We just put negative or zero for the y value there And then I can merge both these guys together So I'm going to select them both the join node and the transform node I'm going to hold down alt on the keyboard Hover over one of the outputs and make a new merge node like so And with that all created I can now run that through a skin node All right, and that will go and create the geometry for me For the sides of my deck All right, and then all we really need to do at this point is create the top top portion there. All right So I can actually just duplicate this join node here. Okay, so just held down alt left click and drag to create a duplicate All right, so I'm going to create a duplicate here And I'm actually going to put this on wrap last to first Like so And currently it's not actually working for me And that's because I have it on only connected Okay, so I have wrap last to first and then I need to turn off the blend And now we have a polygon sitting there and all we need to do is reverse that All right, very cool So with that done I can just pull this guy right down next to that other skin node And we can just merge these guys together. So now I've got A nice little box that sits on the back of the the ramp there Okay, so let's go and take care of the UVs for this All right, and that's pretty easy. Let's make two groups for this. I'm going to say big node here Well actually, you know in this case, it's not really all that necessary We'll make a group here just in a little bit for the whole thing But let's just drop down a UV unwrap and drop that that guy down there and Let's go and take a look at the results of this. All right So we get all of our geometry. I would actually like to have these guys all still connected to each other So that's kind of what I was thinking of at first. So let's make a group node up here. This way we could Unwrap them differently using different methods. All right, so we'll call this the top Like so and then I'm going to duplicate that with the alt left-click technique And this is going to be called the sides Like so. All right, and this works because by default this base group is option is enabled So if I were to take a look at this in the scene view All those polygons are selected Okay, so in this first unwrap node, I'm just going to Select the top. All right, we don't need to do the sides. I want the sides to go connected So that's perfect for the top And then I'm going to use a UV flattened node for the sides. All right So let's wire this guy up like so and then for the group. Let's do the sides All right, let's take a look at this. Yeah, so this basically is unwrapping it nicely So all the faces are connected. All right, so I won't have any seam issues beautiful All right, so then let's do a group delete. I don't need those groups anymore And I don't want them, you know dirtying up my network here So I drop down a group delete and let's use this little drop down here to delete the inner prims the sides and the top We don't need any of those groups anymore And while we're at it, why don't we do that same operation? So I'm going to duplicate that node cut it off like so using the scissors That's the y key on the keyboard. All right And let's just get rid of all the extra Groups that we don't need here. So I'm going to clear this out and just take a look at the groups that I have And I don't need end caps or inner points for that either. All right, so let's duplicate it Let's use y to cut the the wire and put it over here And let's take all the groups that we have so I don't need backs or extrude side there And this is just good to clean it up. So you don't end up with a bajillion groups on your mesh at the very end All right, so let's take a look at all the groups that we need to do here And so I'm just going to get rid of inner points. All right. So now we're all cleaned up. We don't have any extra spare groups just hanging around. All right, so I'm going to grab at the assembly node and just move it down here These guys are this one ended up a little bit long there just to make the the deck All right, so let's go and group this guy. So we'll call this the deck group Like so and actually let's put that lowercase. So I like to keep all my group names with lowercase values And then let's go borrow the color node from the ramp over here because it has that same color And I'm going to cut the wire there And we just want to go and put this at the the end of this deck Network here And there we go. So now it's the same color. Awesome. Then let's put down our null node and we'll say out Tech Easy peasy Well, let's add that to our assembly node So I'm going to create a new slot by hitting this little plus button and then drag and drop it right into that extra slot there Now I've got all this geometry coming together nicely. Let's turn off all of our point numbers and prim numbers Yeah You know, not a complex model, but a really good exercise in learning how to procedurally model one thing I noticed I should actually extend the pipe out a little bit farther than the edge there So to do that up here where we're making the coping All right, and I get that last Curve there Let's go and Make it a little bit longer. So let me actually do that here before we send it out All right. So what I'm going to do is drop down a transform node After that node there and let's just give it a little bit of scale in x. So it's just a little bit bigger So we'll say 1.15 All right, let's take a look at the result of that And that was way too much because I put in 11 Yeah, that's actually still too much. Let's do 1.05 And let's do 2.5. I just need a little bit doesn't he be way off Do 005? Yeah, that's what I'm looking for Okay, cool. We are ready to go. We've got UVs for everything.  I'm a little ocb about it all You don't have to do this All right, Houdini is flexible. You can create a workflow however you see fit All right Cool. So there we go. We have our deck in place We're actually almost done with this whole entire model pretty cool Let's move on to take care of the railing now. Thanks much


If you want to know how to use scenes in Ecamm Live and even which scenes to set up, then this video is for you. Hello and welcome to Take One Tech, my name is Alec. And today we're talking all about scenes in Ecamm Live. And what I've done is I did actually create a four and a half hour mammoth tutorial all about how to go from zero to a full production setup in Ecamm Live and even with Stream Deck as well and building out all of our icons, overlays and things like that. And I'll link to that video up above and also down in the description, because as I say, it is a long one and you may not want to sit through all of that and some of it you may already know or all of it you may already know. But I'm breaking these videos down into smaller, more manageable bite size videos that there are for people basically maybe like you who just wants to watch one particular video at a time or only interested in one part. With all that waffling said, I'm gonna cut now to the video which is where I talk all about different scenes and yeah, I'll join you on the other side and let's talk about scenes. So as I say, I've got all of these scenes. We're not gonna build out all of those but what I'm gonna show you is the sort of building blocks to give you a few different ideas and then you can try this out for yourself and you'll soon figure out how you can make multiple different scenes that might be similar to each other. So I thought we would talk next about actually planning these scenes. So if I come out of demo mode, we can talk about that. And so planning our scenes. The scenes that I thought we would look at first of all is like a sort of a full frame shot if you like a main window, which is something like this, a head-on shot that you might want to use. We'll look at how we can put a nice little border around it as well. I don't tend to have a border on mine but I know a lot of people like that. So in terms of actually building an overlay, I thought it would be a good little exercise to go through and then you can choose whether you want to use it or not. The next one I thought we'd do is a screen sharing overlay and so I've got a screen sharing overlay like this, which is my picture and that happens to be my website but I'm just sharing my other computer screen actually. Then I thought we'd have like a phone cam or a top-down cam. So those could actually be the same thing. I'm using a little old Logitech C920. Here, I can just even pull it into view. There you go. It's my little Logitech camera and I've just sort of repurposed that as a top-down camera. Incidentally, the camera that I'm using for anybody who is interested is a 10-year-old Canon EOS 60D. So not cutting edge by any stretch. It's not a bad camera but it's just not the latest and greatest but I did do a video about how I actually used that and hooked that up for, originally I did it as to improve my quality of my Zoom calls but I've done a video all about that one as well and how to link up like an old DSLR camera to your computer as well. So you can judge for yourself what the picture quality looks like. It's, as I say, 10-year-old camera with the kit lens. So yeah, I think it's probably passable in terms of quality. So yeah, I thought we'd do that camera shot, the top-down camera. And then I'm also gonna show you how you can use your phone as another camera as well. You might wanna use that as a top-down camera if you don't have a second camera or you might want to use it if you want to do something like, well, for example, I'll just quickly show you. So if I want to go to share my phone camera, for example, to give you a little tour of what my studio looks like, which you might want to do, this is quite a common sort of use case. If I click over to that now, there we go. So there you can see, there's my monitor. I've got my old 2013 MacBook sitting over the side and then there you go, there is my Canon camera. That is how you can use your camera as your phone rather as a second camera. It's a great way to repurpose an old phone. So this is an old 5S and yeah, it's getting put to use as a second camera and also as a second or third stream deck. But I'll show you about that later as well. So back to those scenes. We've got the full frame, the screen share, top-down and the phone camera. And then the final one is, if you're gonna be doing live streaming, then you may want one of those sort of the live stream countdown timer. So I'm gonna show you how you can build that scene and build it out with the countdown with some music and things like that and some graphics on there as well. So just in case you're not sure what I'm talking about, this would be something like this one. So yeah, when you're waiting for your stream to start, something like this with some music in the background, your timer and so on. And so yeah, we'll build this one out as well. So let's start from, I muted my own mic there by accident. Let's start from the beginning and I'll share my screen with you again. And then we'll let's build this all out. So let me just hide some of these so that you don't have to see them all. Now yeah, so we'll start with a blank scene. So we just got to here scene, new empty scene. And as a default, it will just use whatever your default camera is. And then in here, you can see that my camera is my EOS 60D, but I could actually select my other Logitech camera that I've got plugged in. And incidentally, if you have got a camera that you're using and you want that to be your default camera, if you've got multiple cameras, then you would just come down here and click select default camera. And that just makes that your default. So whenever you start a new scene, it brings that camera in. Now there are some options with the scenes. Let me just call this one. Got my keyboard, oops, a daisy. A little bit too far out of the way for me to reach here. Move a little bit closer, that'll help. So let's call this scene, just call it main. This is our main scene. And there are a couple of other options with the scene. So you can click in source and you could change it between either a blank scene, the camera, the screen share. And now that's sharing all of my screens by default. But we'll come back to this when we talk about the screen share afterwards, actually. So or you can have it as a video file if you just want a video to play as part of the whole scene. So we'll start with a blank scene or rather a, I say a blank screen. It's not a blank screen, it's a blank scene with just my camera. So what I want to do now is I want to just get rid of this quite glaring background. So as I say, I'll just come up here and I'll toggle on green screen. And then I want to just come into here. It defaults to this sort of brick background that you may have seen in some other demos and things like that, but you can basically choose any file. There are some of that come stock with it. So if you want a beach background personally, I would avoid anything like this, except for the odd joke. Doesn't necessarily look the best, certainly not for meetings or things like that. So then there's different ones like this. In the video about green screen, which I've mentioned that I'll mention again and put the link to in the description, you've got to be careful if you are using green screen, things like this, it looks obviously wrong, doesn't it? And it's all to do with the camera position from my camera is different to the camera position that was used to take the picture behind me. Also, the lighting is very different in terms of tone and things like that. So I explained that all of that in the other video, but yeah, just something to be careful of when you are doing any green screen stuff. So I have actually got this background, which is the one I'm using. So I'll just stick to that, which sort of slightly matches more my, the sort of tones that I've set up in terms of the camera settings and things like that. But the sort of finer detail of that, I'll leave for another video. We're just going to go through now the actual process of setting these scenes up. So that's our main scene. Let's say we also want to have, in fact, let me go back to that list, which ones we were going to set up. So the next one would be a screen sharing scene. So if I come back here now to our new scene, main it was, wasn't it? That was a new one from older. So yeah, the scenes, you can just drag them around if you want to reorder them in the scenes panel. And you can also, which is, I think, quite useful, group them into folders. So if we wanted to have a collection of scenes, in fact, if I just create the next one first, I'll go scene, new empty scene, and then I'll call this one screen share, screen share. In fact, what I'm going to do is I'm just going to call these demo because when we come to the stream deck part of it, I need to be able to identify these to show you how we can pick them. So I'll put them as demo just so that we can be clear about which ones they are. So yes, I can move these around and put this one up at the top as well. And if you did want to group these together, you can come down to the bottom here and you can see this little, you can either add new scenes from here as well or duplicate scenes or you can add a new group of scenes. So I could add in here my scenes. You can call them what you want, but then they can just simply just drag those in. So you can see how I had done that with my collections of scenes that I had before. So I've got them grouped by different use cases and then even within those, you can have folders within folders. If you do have a lot of scenes, it just helps to organize them. But I'll keep them simple for now and keep them out of there. So we've got our demo screen sharing and then we've got our screen, I beg your pardon, I've called that the wrong one. That was the demo main, wasn't it? That really confused me for a minute there. So that's our demo main. That's our demo screen sharing. And I'll just delete this one because that was one I made earlier for some reason. So the screen sharing, we're just gonna go to the source and we're gonna change from camera to screen share. So now we've got our screen share. When you are screen sharing, you can choose between different things. You can either show the specific display. So I've got two displays. And as you can see in here, there's one of them over here and there's one of them over the other side. So you can show a primary display. So that will show whichever is set up as your main display or you can change it to your secondary display. Now, what you can also do is, in fact, if I just move that display over for one second, I'll come to a new thing. In fact, there you can see, I've got my little stream deck panel open for our little demonstration later. What you can do in the screen sharing though, is you can actually select to share, close my preview down. You can actually select to share a specific application. So rather than the secondary display, you could say the current application and that will just share what is open. And what that does is actually, as you move that around, it adjusts the image to make sure that it keeps it always centered and framed in the shot. But for now, I'm gonna actually set that as the full screen. So we'll select the secondary display again. So now as you can see that's showing my full screen over there. I just change over to a different window. You can see that's now got all of my screen in it. You can see also that it's put a picture in picture of my camera in here. I've got my mouse on the wrong screen there. You can see we've got this picture in picture. Just move that preview out of the way. It's got this picture in picture. You can move this around anywhere. And what we want to do is we want to have this screen share as I showed you, well something similar to this, something similar to that is what we're trying to achieve. So what we need to do is we need to have, rather than that screen be full screen, we need it to be a part of a screen. Now, this is the one thing which I either can't see or there isn't a feature, but I can't see how to just add a screen sharing in in a separate window. What you can do is you can swap between these two. So I can make this is now my little window and this is my sort of background. Oh look at that. You can see the top of my screen share. That's really my green screen. It's really broken the illusion now, hasn't it? In fact, let me fix that. You don't wanna be looking at that. I'll put this back to this one. So yeah, you can see that now I've got my screen sharing is my picture in picture, whereas my main camera is the feed that you've got there. So what we're gonna want is we want this, something like that. But what I can't actually make it do as yet, either through my own ineptitude or through a lack of a feature. And I'm thinking it's the latter to be honest, is I want to have that there and I just want to have the camera as a separate pop out window that I can put sort of next to it so that we can replicate something like I showed you in my example there. So the only way that I've been able to find to do this is actually just add the camera in again as a second feed. So if I do that as an overlay, I'm gonna come down to the bottom. My stream deck just swaps straight out of my e-cam live controls there. But down at the bottom here, you can see these overlays I can add in a camera overlay. So I'm gonna click on that one and that's now popped in a camera overlay over the top of that one. And in fact, let's try something a little bit different with this. Let's try and make it so that we've got this one because this is a good opportunity to show you the controls that you have over the camera. So obviously I can select the camera but you can also change the aspect ratio. So that's the dimensions, how wide compared to how tall it is. So if I go from here and I can change to instead of widescreen, you could change it to a more sort of four by three old school TV aspect ratio. Or you can go to square, make it a square or we could have it as tall. So that one, so there you can see it's more like that. That would be a good shot for a mobile shot. In fact, we'll have a look at that afterwards. Or you can make it a round camera, something like that. If you wanted it in the corner, you were doing screen demos, then that's quite a popular one. Or you can actually just make it completely custom. So if you click custom, then basically what you can do is you can just drag it into absolutely any shape you want. So I tell you what, for this one, let's make this as a tall one. And we'll have a view that is, if we ignore the camera in the background, all we'll see is the desktop image from the screen sharing. And then we'll see this one here, maybe make that a little bit smaller. But we'll play around with that when we make the overlay. So it was just to give us an idea in terms of what we actually need to make for our overlays. So I hope that makes sense. Hopefully if it doesn't yet, it will do a little bit later. So we've got our main view and we've got a sort of desktop demo view. And then let's have a look at what the next one we said we were gonna make was a top-down shot. Well, actually a top-down shot, we could have something very similar to that last one that we just did. But let's say we're gonna change some of the shape of it a little bit. So let's come for that one. In fact, let's look at a new way to make out these scenes. So we'll start from here and then let's just go to start with this blank one again. And we'll come down to this plus sign because we're just gonna add a blank screen. And again, it defaults to this green screen. Incidentally, the settings that you create for the camera are specific to that scene. So when it brings in a new scene, it will have like no green screen on it. But if you remember from this one, when I added this camera in as an extra camera, it's remembered the settings of the camera from this scene, which is the way you'd want it really because you might want to have different scenes with different backgrounds. But then if you're adding the camera into the same scene, then the green screen background, you'd probably want to be the same on a scene by scene basis, if that makes sense. So anyway, let's go to a new scene and this is gonna be our top-down scene. So let's say for this, we're gonna start with a blank canvas. Now we want to add in our top-down camera. So let's come down to the bottom for our cameras. And we'll click on that. And then we're gonna select in here that we want this, this is my Logitech. Now, incidentally, when I went and did my NDI source for, sorry, skipping ahead to the method there, but when I did my source for my top-down camera, that my mobile camera that I showed you earlier, that popped up here, and look, that's my wife's old mobile that I've pinched there for the case of this. So if I click my HD webcam, my Logitech webcam rather, this is our top-down shot. And again, you can see it's keeping that scale. And if I drag the corner, but we might want this to be a square. In fact, let's see if we can make this something a little bit different. So we'll make this one perhaps like a top-down cam like that. And then we're gonna want ourself in. And it's interesting, you can see, or useful rather, you can see how it's put that sort of blue bounding box as a sort of guide so that you can see how close it is to the edges. And you can sort of line things up with that. I'm actually gonna put another square camera in here. So we'll come back down to the camera at the bottom, click on that one. And then let's come up here and we'll change this to a square as well. So just sort of having some ideas here about how we can make this work. So you might want to do something where you're doing a side-by-side, where you're doing a demo of your desktop or something like that. So there we go, we'll put that one, put this one here like this. And let's get rid of that background again. I'm just gonna stick with my default background just so that it's consistent. Another thing I should point out actually about the green screen just quickly because it is quite a useful feature is if you have a green screen which is sort of narrow and just covers behind you but your camera is a lot wider, I think I can demonstrate this. If I change my aspect ratio, I'm going off topic a little bit here but I think it's useful to see this because actually my green screen does not go all the way to the top of my frame. So I definitely had it earlier, didn't I? When I was, in fact, let me change this to square. Hopefully this should be quite clear what I'm trying to do when I've finished on mask edges. Ah, it's there anyway. So can you see there's this little white patch at the top? Well, that's basically the top of my green screen. That's where my green screen ends and there's actually the wall behind it. So it's not normally an issue because that happens to be cropped out of frame but if it isn't cropped out of frame for you you might even have one of these sort of small narrow green screens that literally just covers behind you and you've got big expanses of space on either side that are in shot that you want to apply your background to. Well, very cleverly in Ecamm Live, if you click on green screen you can see how it's gone to the edge of the green screen but we've got that gap. Well, you can simply click on this little one here, mask edges and it fills that all in for you. So you don't need to have a green screen that fills your scene in order to have your background fill the entire background. I hope that makes sense. Sorry for that digression but I thought it might be a little bit useful for some of you. So this is back to the square and we want to resize that so that we're sort of the same size as the others. I could actually just put one over the other. Now here's a thing that's worth mentioning. What I was gonna do is put this one over the top of that and just scale it till it's exactly the same size but you can see how it's actually behind each other and that's because anything that you add into the scene is sort of stacked and layered and when you come into your overlays on this side you can see that they're listed in a particular order. Well, that is the order of the layers. So the anything that is at the top is at the top most of the stack of overlays and anything that's further down than that is at the bottom. So if I wanted this image here or this overlay, this camera to be above the other one, I would simply grab it from here and drag it up to the top. So now you can see it's popped to above the other one. So now it's below it and now it is above it. So all I was gonna do here is just to get these so that they're the same size. I thought, well, I'll just pop them into here and then I can drag them out. Whoops, drag them so that they're exactly the same size and now when they're side by side it should look a lot better, a lot neater because they will be the same size. Right, so there we go. So that is now I'm gonna just double click in that one and call that one demo and what's that one, that's top down, isn't it? Top down, okay. Now that camera one, by the way, so we've got, if I come back to these, so we've got our full frame, we've got our screen sharing and we've got our top down. Now we wanna do our phone camera. So maybe a good way to do that would be something similar to this other one. So we want something similar to that but instead of being done in this way, we want to have the camera there and maybe our main shot here. So I'll use this as an example. I've shown you how you can use your camera sort of landscape, sorry, your phone camera landscape as a camera, but for this example, let's just say that we wanna actually use it to demo anything that happens to be on our screen on the phone. So what we'll do here is we'll do a new blank scene again. So click the plus and then we'll call this one demo phone, again, I'm just calling them all demo because we'll be later, we'll be linking our stream deck to this so that we can just click between them as I'm doing. In fact, maybe now it might be a good time to show you exactly what I'm doing just so that it makes sense while I'm talking. I'll go over stream deck a lot more a bit later but this will perhaps just help. So here is what I control my e-cam live with. Actually, I use this one and I've also got this one here and I've got stream deck on a mobile as well. I'm actually using this one to flick between the slides. So where I bring up my slides like if I press this button here, it's just brought me back to this scene. And if I press top down, it's brought me to here and I can activate all my overlays and things like that. So things like, yeah, once again, I could activate my, don't forget to like and subscribe is that button. So when I click that, it comes up. So if you are finding this useful, now might be the time to go and hit the like button or subscribe to the channel and turn on notifications so that you get notified when there are any other videos coming out. And I've also got my animated overlays. So my Buy Me a Coffee link. So that's probably the easiest way to support the channel is to go to buymeacoffee.com slash take one tech. And that's a way that viewers can support the channel if they're finding the content useful. But that's all done from the stream deck. And I'm gonna show you how you can basically do all of this and create these icons and everything for your scenes as well. So yeah, if I come back to the phone shot. So as you can see that before I had the phone as a camera, so if I flick over to my camera, I'm now activating the camera on my little phone. And then when I press the camera button on here, I've got this one that's my mobile. You can see that it is now my camera view. But what you might want to do is you might wanna have this as a vertical. So you would going to do a, so now I've turned my camera around but that doesn't quite fit on the screen, does it? So what I thought we could do here is we could make a, I've just muted myself, by the way. I thought what we could do is we could make a view that was more kind of like a vertical aspect view. So for the camera view, we'll make one that's our picture. And then like if you were gonna show, for example, like an app or something like that, then it would be useful for that. So let's get that done, shall we? If we go over to our demo scene here for the phone and sorry, take my buy me a coffee link off there. If we go to the demo for the phone and what we wanna do for this is let's just build it out from scratch again. So let's do the source as blank. And here we want to have, maybe we can actually duplicate this one. And what we want to do here is change the, this one we want to be our phone source. So let's change that to be our phone. So now you can see I've got my full camera, my full phone picture. And then let's change this one. Let's make the source blank and then we'll just add in just purely the camera. There we go, drag this one out here, something like that. We can play around with the exact sizing and positions of these afterwards. And now you can see that the phone's gone behind it. So we'll drag that one down there. So now the phone is above it. So yeah, it's a bit hard to see this perhaps with it being black on the black background. But once we've got our overlay in place, it'll all make a lot more sense. So finally, which other scene did we have to do? We said we were gonna make a countdown timer, which is something like this as I showed you earlier. So what we need for that is, well, all we've got at the moment is the picture. So let's come and just create another blank scene. I'll come back to this one first and we'll do Daisy. That seems like it's not quite remembered where we are with it, isn't it? There we go. Something's happened to that demo phone one, hasn't it? Whenever mind, let me just move on and we'll come back to that one. If it's still not working later. So we'll come and add a new scene. And this, all we're gonna do for this one is, here we go. So we'll call this demo, count down. Right, and for this one, all we want is, again, we'll start with a blank scene and we'll add a picture. Whoops, I added the wrong thing there. I wanted a, I wanted a camera, not a picture. And let's just make this square for now. We can change it around afterwards, square. And we're gonna actually put an overlay which will cover this entirely anyway. So you'll see how we'll do that after. So let's just add the green screen again to get rid of that background. And so yeah, what we're gonna have here is some sort of big overlay. We're gonna have a little picture in. And the reason, by the way, why people have a picture in on the timer is because it just adds a bit of visual clues to the video, that there's something actually happening rather than you could just have a placeholder. But this is the one that I would normally have and this is what we're going to build out. So we've got the main scene, we've got the screen sharing, top down, the phone and on the side here and the countdown timer. So let's go into the main scene and let's get our overlay in. So now we're gonna come into our overlays panel here. You can see this one, the different types of overlays you've got. We used a camera overlay before, obviously to build out the scenes, but you also can add in a widget. So the widget is actually something like the By Me A Coffee link that I added. So that link there that pops into the bottom, bymeacoffee.com slash take one tech. That's actually an HTML code snippet, I think. That might be not the wrong terminology, it's not actually HTML, but it's a code snippet basically that you add in and yeah, it's a sort of a web widget for a better word. So you can add widgets in, but you can also add in text, a timer, which will come to a movie or an animation, which we'll use as well. But for now, we're just gonna click on this one, new image overlay. So I'll click on that and it should bring up a window to pick out our image overlay. So I've got these in my example files, image overlays and that was the first one is the frame. So I'll click on that and look at that. Isn't that looking better already? In fact, that does actually look quite good. It looks rather in keeping with my background colors. So there we go, we've got our first overlay. We've got an image picture overlay, a picture frame and you'll notice as well, because we sized it correctly in Keynote, by remember first, if you remember when I started the file, I had the aspect ratio set correctly from the start. So it's just automatically bought in at the right size.  So let's have a look at another one now. It's starting to get exciting, isn't it? You can see it's actually looking like a bit more like a production that we're building out. So you can go over to the screen share one. Let me click on the overlay button again, down at the bottom. And now we want to go to the second one, which was this one, click on that. And there we go, that's looking good as well, but we need to resize things a little bit in this one. So let me just come and grab my, whoops, right. So this is a good point to talk about locking. There are different things that you can lock. You can either lock a scene. So if I come over to this scene on this side, we're in the share screen. If I click on that little lock next to it, it doesn't appear until you hover over it, but click on that. And now I can't do anything to the scene. But we do want to make some changes to the scene, but we just don't want to be moving around this overlay because the overlay is in the right place. But you can also lock overlays. So if we come into the overlays, you can see that we've got this. This is the overlay here, overlays in the current scene. We've got the camera, which is this one. And we've got the image overlay for the border. But if I try and drag my camera, it's actually trying to drag the image overlay, I think, or maybe the camera behind it. But if I click on the lock, it'll lock that overlay. In fact, having said that, is that moving something else? That might have been moving the camera behind. But nevertheless, locking the overlay in place is, we definitely need to do that. So now I can drag this camera here and drag it out so that it covers the corner like that. In fact, it's actually slightly different size, isn't it, to this one? But this is a great opportunity to show you how you can click in here. And instead of making that tall, we can just make that a custom size. And so you could literally, no matter what size you've made your overlay, you just drag the corners. As you hover over it, can you see how it's just dragging the corners there? And we can make it exactly the same size as our overlay. And this is what I was talking about. If I was to put that there, you can see how the edge of that corner is just sort of creeping out. That little corner there is just sticking out. I hope that's clear. So all we need to do now is because this border is big enough that it actually will cover it. If I just put that exactly to that line on the inside there, there we go. So that now no longer sticks out over the edge. So yeah, then we want to change this screen share to be that size. Now I've actually made this the size for a standard sort of 1920 by 1080 display resolution. Although here I'm actually sharing my MacBook screen. And so you can see that my MacBook screen is 1920 by 1200. So it doesn't actually feel it properly. So just bear that in mind if you're going to share a MacBook screen that isn't a standard resolution. We get the picture anyway. We've got a nice frame, haven't we? So don't want to dwell on that too much. And let's come down to our top down shot or the, let's do this one that's the same as this first actually. We got this one for the phone, didn't we? So let's come in and add in our overlay to that one as well. Click on image overlay. Here we go. And so I'm going to do the same thing again. I'll just, I'll lock that overlay so that it's not going to move anywhere now that it's in the right place. I'll drag this one over and then I'll grab the, of course sometimes if it's behind a button you don't want to press the button by mistake because I'm actually recording at the same time. In fact, let me just do this the easy way. I'm going to line up one corner where I want it. I'm going to drag this out of this corner. Do you know what? The more I'm looking at this overlay the more I'm thinking, I don't know why I don't have this overlay on my videos. It does seem to match with my color scheme rather well. So there we go. We'll drag that out. So now we've got my phone view. I'd forgotten I was still on phone view. My phone's getting rather hot. So there we go. That's how this scene works. We've now got a little phone camera that we can use or if I wanted to demonstrate my apps I actually took off all the apps on here but I've got my phone set up. That is another scene that we've got built out. So the next one which was a little bit different is our top down. So now we've got an overlay for this one as well obviously come down to the bottom go into image overlays and we've got our side-by-side shots. There we go. We add that one in. Look at that. It's looking professional, isn't it? And I don't know how long we've been filming now an hour and 45 minutes and there's a lot of waffling in there a lot of explanation going on as well but we've built out almost a complete set of scene overlays. So as I say, I will be making complete packs and things like that available further down the line but just give it a go yourself. Take these files that will always be free and play around with them and have a go yourself because it's one of those things that once you get into it it becomes pretty straightforward. The last one is our countdown timer. So we've got a scene for that one, not much to do here. Whoops, I just clicked on a movie file. We don't want that just yet. We're gonna do some animated overlays next in Keynote as well and that is just as easy. So let's come in here and click on this one. Stream starting soon. So our picture is a little bit on the small side. So let me just put this just like that. We could make the image circular if you want. So if I came into this image, I actually made it as a circular image like that. We could do that. And so we could make it the same size that way. But given that we've got this as instant alpha you can do it however you want really. It doesn't matter, does it? Because it's blocking out whatever's in the background anyway. So let's just make that square like that. There we go. Doesn't that look good? It's a pity about the on-screen talent. Nothing I can do about that I'm afraid. I can touch up the colors, but yeah, there's not much I can do about the actual looks. So there we go. We've got that one. One thing that I did on mine, I don't know whether it's worth it or not, it's just a little stylistic thing is I actually changed the camera to be background. But if you're using a green screen you'll know it didn't change the actual background. And so being a little bit, being the way I am, I actually made a black and white version of the background so that it matches as well. But there you go, that's another story. So whether you want that or not, whether you want it to distract from what's going on, it's just another option that you can have. I'll put this all back for now though so that we're in full technicolor. So that is the stream starting soon. But if you remember, my one also, the one that I showed you earlier had a countdown timer and music on it as well. And so the countdown timer is quite simple. It is an overlay in Ecam. They make that very easy as they do with most things. If you click on the new countdown overlay and here you can see you've got the countdown. You can choose your typeface and the size and so on. And then you can also add like different colors and effects and things like that. Let's just start with this for now. And just let's get our positioning right first before we think about it. I mean, we could have it there. We could have it up in this top corner. We could have it anywhere. Let's just for the sake of argument, let's make it nice and big and bold here and let's try and match this color here just so that I can show you a couple of extra things that you can do with it. So let's click on that again to edit it. Now we've got the duration here. I'll come to that one in a minute but we can change the size of it. So let's make it nice and big. There we go, 144. And the color of it as well, we can change the color. So let's make it something like, in fact, let me just have a look what color that one is. It's this blue color. So you've got nice, nice big bold writing there. So we wanna make it something like that blue. So let's just come, we've got the color wheel here and if I just change this, we can change it to some nice, nice blue color. And then let's say we want to add a background to it or we could add a shadow to it. It's got the shadow already selected. So now let's click save. So now we've got our countdown there. So in terms of the actual timing of it, one thing that I have with my countdown timer is I have it so that it plays the countdown but then there is this function as you can see if I go back into edit, what do you want to happen when the countdown finishes? So what we want to do is have this set up so that when we're waiting for our live stream, we've started the stream but the actual official start hasn't happened yet so it's counting down to it. You could toggle this button here, which is, or click this little toggle here, which is go to the next scene when finished. And that will mean once it's counted down to zero, it'll just immediately go into the next scene. So let's click on save. And in order to do that, what you want to also then make sure is that you move this demo up above your main scene or whichever scene you want to come after it. So here it would finish this countdown and then it will go to the next scene, which is our main scene. Okay, which is that one. Now the next thing you might want to do is add in some music. Now I've got a particular piece of music that I use in my countdown timer, which is somewhere in here, this one, is it? No, it's not that one. This one, stream starting music. So it's this piece of music. As I said before, all of the music, just to remind you once again, is from Epidemic Sounds. So there's all sorts of different sounds, different genres of music in there. They've got a huge catalog of, you know, good quality music. This may not be to your taste, what I've got playing on at the moment, but there's something for everybody in there as well. And like I say, it's the easy way to make sure you don't fall foul of any copyright laws or anything like that. And yet, go to takeonetech.io slash epidemic to get your free trial of that. But in any case, I downloaded this one that's sort of happily playing away in the background at the moment. Let me come back to our demo countdown. And so what I want to do is I want to add that as part of this scene. So what I can do is, if you just bear with me one second, well, that might have given the game away a little bit there because the timer has suddenly leapt to a different time. And the reason was for one minute then I forgot how to actually add the bit of music to the current scene. So I just had to check on it. So yeah, we don't remember everything all of the time, do we?  And now this has appeared in the current scene. Now I happen to know that this piece of music is, I say I happen to know, I haven't written it down, three minutes and 25 seconds long. So if I just pause this for a moment and I go to the timer, then what I can do is click in Edit. And I make this time here just the length of that track that I've got. So if I go three, 25, this scene is basically gonna last exactly the same amount of time as that piece of music is. And so when we run the scene, the music will run. And then also when the music is finished, is the same time as the timer's going to finish. And so then it will advance to that next scene. So that's how we set up that. So the next thing that we're going to look at is we've done now basically the picture overlays. We've added some music in, sorry, I shall just actually show you exactly the process for how to add that music in. If we click on the Add button in Sound Effects, it's quite simple and you just go and find a bit of music and click on OK. And then it will just add it into this section here. So the next thing that we're going to look at then is from here, we've done the, as I say, the picture overlays, the timer overlay. We've added a camera, we've added a new group. I showed you how to do that before with the scenes. That's the same with here. If you want to group your overlays together, like I've got different overlays for various different things here. And then you could add some text as well. This is actually something that's worth mentioning now because in fact, we might want to add this to this stream. We've got this background, which is stream starting soon. But what you may want to do is on a sort of stream by stream basis, actually update the titles. So we can do that by just adding some text so we could have like whatever today's stream is, whatever your title is. So we just click on add. And so you can, whoops, I've added that into the wrong place. For some reason, I've switched over to completely the wrong scene there. Let me just come back to that a minute. There we go, that's a bit better. Let me just pause that timer. You see how the music started though when we came back to this scene because it's programmed to start as soon as the scene starts. So, yeah, but what I was trying to do having added it to the wrong scene is add some text and we called this our, let's call it stream title. Stream title, there we go. We've got our stream title. And let's say we want to match the typeface and font of that one. We'll click into here and then we were in Helvetica before. So let's just change the color on that. So we'll pick white, come in here and pick some random colors or they've got some fixed ones in here. So stream title and we'll save that as that. It's a bit on the large side, that isn't it? There we go, stream title. So if the stuff you want to add in on a sort of stream by stream basis that's not the same for everyone, like obviously stream starting soon, then you can do that in here. There we go. I hope you found that useful and if you did then don't forget to go down and like and subscribe and turn on those notifications so that you get alerted any time I make any new content. And also don't forget that all of the icon packs and templates and things like that that I've talked about in these videos are available as a free download from my website. So simply head on over to takeonetech.io and there you'll be able to see at the top of the page there is a link to the actual full tutorial that this is an extract from. But also just below that you'll see that there is a area where you can download that full icon pack and template pack and make it your own. So yes, that's the place to go. And also that's the place to go if you want to contact me as well. That's the easiest place. There's a contact page on that website but also you'll notice just down in the bottom right hand corner there's also a little chat box. You can always get in touch with me that way. Incidentally, if you're not already using it then this whole video has been produced with Ecamm Live and it's the my video editor, my video production environment should I say of choice certainly better than what I was using before, yes. And so yeah, it's highly recommended. It certainly made my life a lot easier. And yeah, if you head over to takeonetech.io slash Ecamm then there you can get a free trial and try it out for yourself and see what all the fuss is about. Well, that's about all for now. But as I said, this is actually an extract from a much larger four and a half hour tutorial that I did, which will take you from zero to a full production setup in Ecamm Live including obviously all of the overlays and transitions and things like that and linking it to your stream deck and designing all of your stream deck icons and all of that business, everything that you need to get going with your video production and streaming. So I'll link to that video down below as well or you can just go and find it on the top of my web page as well. But for now, that's all from me and I'll link to some of the other videos from that tutorial also to the right hand side. And until the next video, have an absolutely wonderful day.


Put it on my Instagram, the link is down below, and you can click on the link below. The minutes have started, guys. Yeah, man.


A lot of our students are first-generation, come from obviously very diverse and different backgrounds. So unless we take the time not just to educate ourselves but beyond that to embrace the diversities and really create relationships, that's the key to our success. We have to balance the rigor in the academic side with the compassion and the love and the caring side of teaching, learning and being a good college. What I'd like to say about that is we really want to normalize health and say that it's okay if we have to provide health. And finally, academic, social and emotional assistance. When people come forward and they ask for those things, it should not be stigmatized because in the end, success with support is still success. This summer, the college enhanced its efforts to emphasize these priorities including a new office of diversity, equity and inclusion and compliance. Which aligns student and employee initiatives.  The Working against One initiative is supportive of WCTC's diversity, equity and inclusion efforts, which are core institutional values of our college. We are incredibly excited to move forward in this work.  And so we care about you as an individual no matter what your story, no matter what your background, no matter what your diversity. So please know that we'd love to hear more about what your story is. And we want to help you navigate whatever it needs, whatever we need to do or whatever we can help you do to make sure that you're successful. Not only here as a student, but successful in a personal setting at home to make sure that you are successful and we're able to help you achieve your goals. It is part of our culture to be relentless advocates for our students and to build positive work environments for our faculty and staff. By developing and expanding our understanding in a systematic and collaborative way, I believe we can improve our services and reach more people, expanding our reach and broadening opportunities for success in underrepresented populations. As we move forward, it's our job to make sure that everyone who wants to can find a pathway to success in the Wisconsin Technical College system. We are all to an extent products of what our experiences have taught us. This is a time when we can and should reflect on those experiences and add them. This deep and shared understanding can make our institutions even better. This is a great time to be a part of the Wisconsin Technical College system and I appreciate your commitment to the WTCS Working as One initiative. Thank you.


We hit the point where everyone everyone in this entire Commonwealth has been touched, whether that's losing a friend, a family member, a neighbor or being worried about losing that friend, family member or neighbor. This addiction crisis reaches every neighborhood of every county across this state. Well, you know, the 37th Psalm says, depart from evil and do good and you'll abide in him for the Lord loves justice. Using Oxycontin money for recovery and just the right thing to do. It's a little bit of justice and so I couldn't be prouder of using these settlement funds in this way. It's the right thing to do. It's taking something that plagued this region, our whole state, and it's doing some good with it. It's helping to save some lives from what ultimately put them in harm's way.


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we were considering the real valued functions and the types of discontinuity. So, we shall continue with that discussion. Let us recall that we had tried following the type discontinuity. So, for this we had considered a function f defined on open interval a, b to r and x some element a, b and we had said that x is a discontinuity of let us say type 1 or another name is jump discontinuity or also called simple discontinuity. So, this means that the limit left hand limit and the right hand limit both exist and their values are different.  So, we can simply say this means f x minus or f x plus does not exist. Before proceeding further let us see some examples of this kind of discontinuities. Again let us take the well known examples which we have considered earlier several times. For example, let us take f of x as integral part of x we can take any interval in fact we can take f defined from r to r. So, suppose I take let us say x is equal to 0 then we know that near 0 again let us remind remember that integral part of x is the greatest integer not greater than x. So, in this interval minus 1 to 0 its value will be 0 integral part of x will be 0 and in the interval 0 to 1 its value will be 1 and. So, it is clear that if you take f of let us say in this case f of 0 minus if you take any number here its value is value of f is going to be 0 and hence it is a constant function here. So, this limit will be 0 and on the other hand f of 0 plus for that we are considering the interval 0 to for whatever delta you take. So, integral 0 less than t less than delta whenever delta is less than 1 the function is going to be constant and its value is going to be 1. So, the right hand limit here is 1. So, this is a discontinuity of type 1 and you can now see why it is called jump discontinuity because the function takes a jump here. Of course, in this particular case it so happens that the value of the function at 0 it is same as this value of the but that may or may not be the case the value may lie somewhere in between also in in general. So, this is the discontinuity of type 1 or jump discontinuity alright. Let us take another example that is also something we have seen earlier let us say g of x is equal to sin 1 by x of course, this will not define the function when x is 0. So, let us give some while strictly speaking it is not necessary to define the value of the function x equal to 0 if I consider in the ring limit left hand limit or right hand limit. But it is since we are talking about the types of continuities let us take some value suppose I take the value 0 this value really does not matter. Now, in case of this you will see that neither g of x plus nor g of x minus exist that is the neither the right hand limit exist nor the left hand limit exist. Now, how does one show that I have said earlier also that the best way to show that the limit does not exist is that you construct two sequences both converging to 0 and limit of g let us say suppose you call those sequences x n and y n in such a way that x n and y n both converge to 0 in this case and g x n and g y n converge to different only thing is that if we want now we have to restrict this sequences x n and y n both either to the right of 0 if we consider the right hand limit or both to the left of 0 if we are considering the left hand limit. Let us take one of those things suppose we are considering right hand limit then in fact even this is also something I think we have seen earlier for example suppose you take x n as let us say 1 by n pi 1 by n pi and suppose I take n is equal to 1 2 3 etcetera then all x n are bigger than or equal to 0 all x n so this sequence lies on the right of 0 and what about g of x n g of x n is sin sin n pi so g of x n is sin n pi so that is 0 and in a similar way and of course x n tends to 0 x n tends to 0 and g of x n also tends to 0 g of x n also tends to 0 but what we can also do is that I can take let us say another sequence y n as 1 by let us say 2 n plus 1 let us pi by 2 then this also tends to 0 and g of pi n that will be sin 2 n plus 1 pi by 2 so that will be 1 that will be 1 so that is 1 for all so x n and y n both tend to 0 and g x n tends to 0 g y n tends to 1 and remember x n and y n both go to 0 from right so this shows that the right hand limit does not exist so this shows that g is 0 plus does not exist and so this is a discontinuity of type 2 of course we can also show that g of 0 minus also does not exist by similarly constructing two sequences but if the objective is to show that it is a discontinuity of type 2 it is not essential if you show one of these does not exist that is enough to conclude that the discontinuity is of type 2 in this particular case in fact both do not exist that is also recall another example which we had seen again in several context namely the so called Lebesgue function it is h x is equal to 1 if x is rational and 0 otherwise can be also described as the characteristic function of q in case of this function you will see that whatever x you take whatever x you take if you take any interval say x minus delta to x plus delta if you take any interval like this then any such interval will contain rationals as well as rational so what you can always do is that you can construct a sequence regardless of what whether x is rational or not you can construct a sequence of rationals converging to x in that case h of x n will converge suppose you take a sequence x n such that each x n is rational and x n converges to x then h of x n will converge to 1 so h of x n will converge to 1 in a similar way you can also construct a sequence let us say y n of irrational converging to x in which case h of y n will be 0 so that means this will have a discontinuity of type 2 for every x so h x is h x has a discontinuity of type 2 at all x for all x in case of this functions we have shown that the discontinuity of type 2 is at x equal to 0 for every other x it is continuous this function sin 1 by x for every x not equal to 0 it is a continuous function at x equal to 0 it is discontinuous and that is of type 2 whereas in case of this function at all integral values it has a discontinuity of type 1 at all other values it is continuous again whereas in case of this function it is discontinuous everywhere and has a type 2 discontinuity all right now we shall consider a class of functions again this class is something that you are familiar with what are called monotonic functions and we will see that as far as this types of discontinuities are concerned monotonic functions have some very interesting property namely that monotonic functions do not have this kind of discontinuity yeah for example we have seen this example integral part of x which I had shown just now that is how monotonically increasing function that is a monotonic function and we have seen that only discontinuity of that is discontinuity of type 1 what we shall show is that something like this is true for all monotonic functions so let us begin by recalling the definition of a monotonic function again it is convenient to define all these things in an open interval so let us start with so f from a b to r is said to be let us say monotonically increasing if suppose you take two numbers x and y in a b and x then x less than y this should imply f x less than or equal to f y we have defined what is meant by monotonically increasing sequence sequence is special case of functions so it is similar and we have seen that increasing monotonically increasing sequences have some special properties as far as convergence etc is concerned so similar thing is true for monotonically increasing functions when it comes to limits and continuity of course we can similarly define what is meant by monotonically decreasing function only thing is that this inequality will be reversed this will become f x bigger than or equal to f y so I can simply write it here decreasing so that will this will be f x bigger than or equal to f y and similarly we shall say it is strictly monotonically increasing if this inequality is strict that is f x less than f y and similarly strictly monotonically decreasing then finally what is meant by a monotonic function monotonic function means it is either monotonically increasing or monotonically decreasing so monotonic this means monotonically increasing or monotonically decreasing well you know several examples of monotonically increasing or decreasing functions or the functions which are not monotonically increasing or decreasing so we shall not go into that already we have seen like one example namely that integral part of x that is an example of a monotonically increasing function so let us go to the paint result what we want to prove is that if a function is monotonic that is the ultimate idea we will need some preparation for that what we need want to prove is if a function is monotonic then it cannot have the discontinuity of type 2 so let us start let us begin with a monotonically increasing function so let us say let f from a b to r be monotonically increasing then what we want to show is that it cannot have this discontinuity of type 2 and to show that it is enough to show that for every x both these limits exist suppose we show that that is enough right suppose we show that that is that is enough so then this is it for every x in a b f of x minus and f of x plus exist in fact we can say something more we can also say what those left and right hand limits will be what we can say is that this left let us say this is the interval a b and suppose this is x somewhere then if you take the set of all values of functions between a to x and take their supremum then that will be left hand limit of x that will be left hand limit of x that is in other words what we can say this l u b of this let us say f of t where t is between a less than t less than x we can say in fact this is what should happen this is f of x minus this f of x that is left hand limit will be supremum of all values of taken over these values of f t over this and similarly f of f of x this will be always less than or equal to f of x and f of x will be always less than or equal to the right hand limit f of x plus and what will be f of x plus for f of x plus you take this interval x to b and take all possible values of f on that interval and take their infimum take the infimum that will be the right hand limit. So, g l b that is infimum of f t where t varies between x and x less than t less than b and we can say something more if we take say two points x and y with x less than y then the right hand limit of x we are not at right hand limit as well as left hand limit exist right hand limit of x will be less than or equal to left hand limit at y. So, also we can say that also for a less than x less than y less than b f of x plus is less than or equal to f of y minus and I have stated this for broadly increasing function similar theorem for a broadly decreasing functions for decreasing functions what will happen is that this l u b etcetera that will be interchanged. So, you can suitably write the theorem for decreasing functions I will not write the complete statement. So, I will say similar statement for monotonically decreasing function. So, let us go to the proof of this. Let us look at this set first and why this is why the supremum of this should be same as your f of x minus. So, let me give some name to this set suppose I call this set a let a be equal to set of all these values f of t such that t varies over a less than t less than x. Obviously, a is non-empty you can always there is some t between a and x. So, that f of t lies here is it also bounded above because for every f t f t t is less than x. So, f t is less than or equal to f x. So, f x is an upper bound for this. So, a is non-empty and a is non-empty and bounded above by f x. So, l u b of a exists it is a non-empty set which is bounded above.   So, least upper bound will be less than or equal to f x.  We already shown that l is less than or equal to f x. Now, to show that is a limit let us use the usual epsilon delta definition. So, let us say that let epsilon be bigger than 0. Then we make an usual argument if you take any epsilon bigger than 0 then l minus epsilon is not an upper bound l minus epsilon is not the upper bound of this set because l is the least upper bound. So, there should exist some element here which is bigger than l minus epsilon. So, let us call let me call the element t naught.  Now, what is to be done after this is clear see f is a automatically increasing function. Remember what is what is the situation here that is we have this interval a to b x is let us say one of the points here and t naught is somewhere here between a and x naught.  Now, is it clear that if you take any points say t naught to x any point lying between t naught and x then f of t will be bigger than or equal to f of t naught because f is automatically increasing and hence bigger than l minus epsilon. So, suppose I take this distance as delta suppose I take this distance as delta then for all then for that particular delta for all t lying between x minus delta to x we will have f of t bigger than or equal to f of t naught and hence bigger than l minus epsilon and obviously less than or equal to l. So, we can say that. So, let us just complete this.  So, what we have is that l minus epsilon is less than f of t naught and that is less than or equal to f of t l minus epsilon is less than f of t naught and see t is somewhere here between t naught to x.  So, f of t is less than or equal to l, which will be anyway less than l plus epsilon. So, what did we show?  Now, this is same as saying that l is the left hand limit of f t s t goes to x. So, this proves that therefore, l is equal to f of x plus. So, what did we prove? We proved this equation that is this is what we had called l and now we have put it l is equal to f of x minus that is left hand limit. We have put l is equal to f of x minus and we have already shown that l is less than or equal to f x. So, all this we have proved. Now, what I would say is that similarly you can prove this other inequality. Just as here I have taken the set a as f t a less than t less than x, you can take the set b as say f t where x less than t less than b and take its infimum and show that that is nothing but the right hand limit. In the same way for example, if you take let us say m as the greatest lower bound of that, consider m plus epsilon that will not be the lower bound. So, there will be similarly some t naught and since the ideas are more or less very similar, I will not go into the details of that proof. So, prove that part yourself show that f of x plus is same as this number and then that number is bigger than or equal to f x. Now, what remains this last thing we need to show that if x is less than y then f of x plus is less than or equal to f of y minus. So, now consider this let a less than x less than y less than b. Of course, this proof is more or less trivial if we just look at the earlier it follows from what we have proved earlier. See if x is less than y that is let us say x is less than y here, I can consider some number t lying between those two. I can consider some number t lying between those two. So, consider x less than t less than y then remember one thing.    So, is it clear from this that see t is bigger than x this t is bigger than x.  So, we have proved all the things required in this theorem. So, in particular we can say that what follows from this is the I will continue here is a corollary. Pornotonic functions cannot have discontinuity of type 2 that is the statement. Pornotonic functions cannot have discontinuity of type 2 which is same as saying that all the discontinuities must be of type 1 that is they must be jump discontinuities or symbol discontinuities. You can say that there is nothing new here it does not have discontinuity of type 2 means what the left hand limited right hand limit always exists and that is what we have proved that is what we have proved for every x left hand and right hand limit always exists. We can say something more we can say that if it is sub-rolonic function and if you look at this set of all discontinuities of f then that set cannot be very big that is that set has to be countable. Again you if you look at the function integral part of x what the discontinuity is the integral part of x it is a set of all integers that is a countable set that is a countable set. So, if you take any monotonic function if you and if you take the set of all discontinuities of that function then that set must be countable set which will also follow from what we have proved just now. So, let us write that as a theorem let f from a b to r be monotonic again here monotonic means monotonic increasing or decreasing then and let us say that let e be the set of all discontinuities of f. So, let us write the full form. So, it e is the set of all x in a b such that f is discontinuous at x then e is countable that is what we want to say. So, this shows that the set of all discontinuities of a monotonic function is a countable set usually in real analysis countable sets are considered small sets. So, this is explained by saying that the set of discontinuities of a monotonic function cannot be a very big set let us take the proof of this. For example, you cannot find a monotonic function whose set of discontinuities coincides with counter set that is clear or you similarly you cannot find a monotonic function which is discontinuous everywhere that is not possible. You cannot find a monotonic function which is discontinuous everywhere in some interval that is not possible because the set of discontinuities has to be a countable set. We want to prove something for a monotonic functions and as we know monotonic means monotonically increasing or monotonically decreasing. So, let us take one of the cases other will be similar. So, as usual we shall say that without loss of generality without loss of generality assume f is monotonically increasing and then let us say that e is the set defined like this. So, we want to show that e is countable let us take some x in e let us take some x in e as x in e means in particular it is also in this interval a b. Then we know that f of x minus and f of x plus exists that is there, but it is a point of discontinuity. So, they cannot be the same f of x minus and f of x plus cannot be the same if they are not the same then either one is less than or equal to one is strictly less than the other which is which should be less than which we know that f of x minus is always less than or equal to f of x plus. So, if the function is discontinuous at x then f of x minus plus b strictly less than f of x plus. That is clear then f of x minus must be strictly less than f of x plus. Now, we do the usual trick here. See these are two real numbers we do not know what kind of real numbers are there, but these are two distinct real numbers and we already shown that between any two distinct real numbers there exists a rational number. So, we choose one such rational number since it is going to there will be several choose any one of them and since it is going to depend on x I shall call that number r suffix x I shall just call that number r suffix x.  Now, after that what is to be done is clear. So, suppose I collect all such rational numbers for each x I have one rational number for each x I have one rational number. So, suppose I take this set of all r suffix x the x belongs to e this is subset of rational numbers and hence it is countable. Hence it is countable this is countable remember for each x we have chosen one rational number and obviously that set is going to be countable. Now, does it say immediately that e is countable we are not shown that for example, it can happen that two different x may correspond to the same rational numbers. See this function x going to r x if we show that that is one one then that is fine if we show that x going to r x that is one one then it will be there will be a bisection between e and this set and then this is countable and hence e is also countable. But that we are not shown right now what we only know is that this map x going to r x is on to because we are collected only those numbers here. Now, let us give some name to this set suppose I call this set let us say f then there is a map which goes from e to f namely x going to r x that is x x going to r suffix x this is a map which takes e to f we know that f is countable. But we know that this map is on to but that does not say that e is countable that will not say that e is countable. But if we show that it is one one then it will mean that it is bisection and if f is countable that will also give that e is countable. Now, what is required to show that this map is one one we must show that if you take two different numbers x and y then the corresponding r x and r y are different. Suppose you take two different numbers x and y then corresponding r suffix x and r suffix y are different. So, for that we are going to use this result next part of this. So, suppose you take two different x and y so in particular we can assume that x is less than y then let us say that. So, to show that to show this part let us say that let x y be in e and x not equal to y obviously if x not equal to y either x is less than y or y is less than x it does not matter which one we call x or which one we call y.  What can we say about r x or r suffix x?  So, r suffix y must be strictly bigger than f of y minus. So, now, can r suffix x and r suffix y be equal? So, what we have shown that if x is less than y r suffix x is strictly less than r suffix y. So, that shows that this map which takes x to r suffix x is 1 1 it is already on 2.  So, e is a countable set. So, again to take the summary of this monotonic functions what we have shown is that monotonic functions cannot have discontinuity of type 2 all the discontinuities of r of type 1 and the set at which the function is discontinuous that set must be countable. Now, let us also discuss one more thing which something that we have discussed about the sequences, but we have not discussed about the functions. In case of functions we have seen that after defining the limits or convergence of a sequence and discussing various properties we extended those things to extended real number systems and we talked about when do we set a sequence goes to infinity or minus infinity etcetera. Now, similar things we will we also want to do for the functions that is see till now we have talked about this what is meant by saying that limit of f x as x tends to a is equal to l and of course we have done it in case of any matrix space and a function f goes from x to y also, but let us now say that f is a real valued function and it is defined on some interval on the real line and it is a real valued function. So, we have we have no problem in saying when a and l both are are both are real numbers. What we want to do next is that we want to consider whether when a or l or both can be extended real numbers that means whether l is plus infinity or minus infinity or similarly a is plus infinity or minus infinity and the way we are going to do it is as follows. Let us first recall what is the meaning of this let us let me let me recall what we have said is this that is for given any epsilon because we said there exists delta bigger than 0 such that such that for all y lying between x minus.  We have seen another interpretation of this or different way of writing the same thing saying that mod x minus a is less than delta it is same as saying that x lies in the interval a minus delta to a plus delta that is let us say x lies in the interval a minus delta to a plus delta. Suppose, I call this interval as v then v is a open interval containing a or in other words it is an open neighborhood of a v is an open neighborhood of a.  So, it is an open set containing l.  So, saying that whenever x is in v f x is in u that is same as saying that f of v is in u. So, I can recast this whole definition in terms of this neighborhoods. So, what I can say is that for every epsilon bigger than 0 there exist delta etcetera what I will say is that for every neighborhood of l for every neighborhood open neighborhood let us say u of l there exist open neighborhood v of a such that f of v is containing u is that clear. So, I will say that instead of this definition I will say that this means for every open neighborhood u of l there exist an open neighborhood v of a such that f of v is containing u this is the meaning of this limit of f x. Basically, we have just written the same definition using the idea of open sets or open neighborhoods. Now, why did I do that? I want to extend this to the case when a and l can be in r star. What is r star? r star is r union these two symbols minus infinity and plus infinity with the usual operations which we have talked about earlier. Now, if I want to extend this definition what I will do is I will just use this definition. I should only say what is made by neighborhood of plus infinity and neighborhood of minus infinity. If I say that then the same definition will go and that is fairly easy. I will say that if you take any interval of this form m to infinity take any real number m take any interval this form m to infinity then that is called a neighborhood of plus infinity. That is what is this set? This is the set of all real numbers such that set of all x in r such that x is bigger than m. We say that this is a neighborhood of infinity. Similarly, if I take minus infinity to m that is considered neighborhood of minus infinity. Let me use some other register of m. So, suppose I take say minus infinity to l minus infinity to l. So, this is the what is the meaning of that this is the set of all real numbers such that x is less than l. This we regard as neighborhood of minus infinity. Now, once we fix that this is what we mean by neighborhood then we just use exactly the same definition. Now, a and l or both can be either it can be real number or the plus infinity of a and l both now can belong to the extended real like that means one of them can be plus or minus infinity or both can be plus or minus infinity. All that you do is that just use this definition and understand that neighborhood of plus infinity means intervals like this and neighborhood of minus infinity means intervals like this. Let us just take one illustration how this definition will translate into the usual definitions that you are familiar with. So, for example, let us say a is a real number but l is plus infinity that is. So, suppose let us say I want to say what is defined by this limit of f x as x tends to a it is infinity. So, a is a real number and l is plus infinity.  Now, but what are the neighborhoods of infinity we have seen the neighborhoods of infinity of this state. So, saying that for every neighborhood u of infinity which same as saying that for every m for every m there should exist that is this is same as saying that for every m there should exist v is a neighborhood of a and if a is a real number neighborhoods are of the form a minus delta to a plus delta.  So, what we want to say that f of v is contending u that means whenever x is in this neighborhood f x is in this. So, f of v is contending u that is same as saying that whenever mod x minus a is less than delta this implies f x is bigger than m. That means given any large number any number m large or small does not matter you can always find a delta such that whenever mod x minus a is less than delta f x is bigger than m this is the meaning of saying that limit of f x as x tends to a is infinity. Similarly, you can take all possible combinations either l is infinity x a is a real number or l is minus infinity and a is a real number or a is infinity and l is a real number etcetera or both are plus or minus infinity. You can write the definition just take this as a basic definition and use this that neighborhood of plus infinity is interval m to infinity neighborhood of minus infinity is interval minus infinity to l and then just translate all those definitions in this form you will get all the usual definitions in usual way in which this kind of things are written in analysis books. We will stop with that.