Tom here from Lawrence Systems and FreeNAS 11.3 U1 the first update in FreeNAS has been released and I figured well Let's talk about that as I have a couple little things I found So if you want to learn more about me and my company hover to LawrenceSystems.com If you'd like to hire for a project there's a hires button up at the top if you want to support the channel in other ways There's affiliate links down below to get you discounts and deals and services that we have to offer Or the other companies have to offer lots of discounts and deals. Anyways, that helps us out with those affiliate links FreeNAS So 11.3 came out and not too long after we have 11.3 U1 and the major thing of course Security is of the utmost concern. So there is a CVE 2019 14907 That has been addressed in this so if you care about this or this problem affects you if Samba is set with the law level three above then the string obtained from the client after a failed character Conversion is printed such streams can be provided during until as we authentication if the Samba ADDC in particular this may cause a long-lived process such as an RPC or terminate in the file server Most likely target operates per process client. So we crashed there is harmless But it's so crashes and this is a problem. Do not set log level three in production was the work around These are probably not things you worry about but I like to mention that they did get patched What they did do is a lot a little bug fixes here and it's all Mostly minor little VM net error on certain settings. It won't unlock with just a recovery key I'm trying like a pool of the pastries I had previously made recovery key to confirm that both path region recovery key I'm testing the ability to unlock the pool in both passphrase and recovery key and you get this I actually had tested this and it looked like you could drop the files in manually at work But it wasn't working through the UI so this like I said There's a lot of little bugs that got fixed on here But the ones that I'm seeing and I've seen people talking about this was this is my free NAS here at the office It's running 11 3 u1 completely up to date and we're gonna go over to the jails and The jails had no problem as you can see here going to 11 3p 6 and then we have my free NAS mini 3.0 e at home purple NAS dot local and it's also same version But the jails the only jail that updated and This didn't update because the system updated was this one It would give a weird error every time you try to update it would just give me a the letter B And I believe I still get that on these if I try to manually update them So I'm gonna do some work to reproduce the error and probably file a bug report or try to dig in further Now it did update from the command line and I gonna get deeper into this in a video But you can update your IO cage by going update Well first you have the fetch and I did fetch the latest 11.3 release, which is the latest Update for there. So if we go IO cage fetch It will grab the latest release it'll pull it and you can do a dash R for upgrade To be able to push the upgrade to one of the jails And I thought that was interesting that I had to do it on one but not the other one updated Find any other didn't I didn't have any error message when I updated and I do see other people talking about that Now I'm not gonna dive too deep into that topic right this moment But I will admit that this was something of note that I ran into when doing the update Other than that everything else with the system seems to be perfectly fine and all the other when you dig through some of the Arata changes and improvements. It's all really mostly all minor stuff on here Create local users on true nas alpha 2 fixed little typos and dialogues that kept opening and things like that And there was an error apparently in the way it was creating the ice fuzzy So that was also part of the problem and the ability to get statuses of the drives if they were resilvering I never have any drives go bad from when I loaded 11.3 even the beta all the way till now I still all my drives are perfectly fine So I haven't done any of the driver replacement, but they have fixed a bug that was causing that so my overall I'd still say to load it But if you are running into that issue do look up how to manually update jails Currently, I do not have a video on how to do that yet. I'm working on I want to do an entire series on IO cage and kind of getting into the nitty-gritty of it I did one on VLANs and apparently they did fix a bug which I didn't see when I was doing my VLAN video But I guess there are certain circumstances and I think someone had commented on that particular YouTube video of that Oh, if they selected one VLAN it would select another one and there's a fix for that in 11.3 So I don't really have any reasons not to load 11.3 U1 But I did note that and obviously the jail updates I don't always pay attention to them as is much will get my home because I'm not running that much in the jails And if it's it's a version behind it's not a big deal because none of it's publicly exposed It's just running my Plex server but Apparently there are some people having some update issues with it my one where I do run things in my business at home where I Focus more on that it updated fine So the ones at the office had no problems no errors updated And it's only the ones that well the one server I have at home the one free NAS mini that had the update for the jail other than that everything else seems to be working fine I loaded it several days ago when it came out I got right on and updated it and want to let it run for a while and I haven't found any Ill effects from it which I think is what a lot of people ask is there any problems you experienced? No, the services I use we still watch stuff on Plex the transmission still works my one Plug-in that is the jail I have I'm gonna go back over to jails real quick My Alcatraz jail, which is my one little home server that works perfectly fine I use it sometimes when I need a different perspective to SS agent something at home Just for random personal projects, but all these things sink thing included all work fine I granted they're still running the old release, but they didn't break I also have the sink thing jail set up manually, so I'm running it not as a plug-in, but as a matter of fact When you run these if you build them manually, and this is what I'm gonna be working on a video on I don't rely on The updates from IAC systems I can actually update sink thing manually by creating a manual jail to run it in So that's why it's a little bit less concerning because the same thing is something I care about a lot And I'm gonna be doing a new video on that as well because it is definitely improved a lot They've had a lot of features over time But it doesn't seem to have any problem running in a slightly older release of the jail But I'm gonna manually update these I was hoping when they first came out because I did see that the update Didn't happen a couple days ago I thought maybe I'll just try and run the update again or try to run the actual Function update and it it just kind of doesn't do so if we go over here And we'll try it right now to show you the error message I get so if we go over here to update It does say updating jail But it doesn't necessarily update it it matter of fact it keeps pulling the same older release instead of the new release And right there's the error branch 11 three release does not exist But it actually does exist and I downloaded it So if you can see that error message, I'm getting right there like so I'm still digging into that but my overall with it It works great. I haven't any problems with this minor exception, but updating from the command line seems to kind of Get around that without a problem as far as the updates So I just have to change the release on there But it'll be doing some further updates and videos on this by the way this mini e does work great But now one thing of note it is a lot slower compared to running it on my other server, which is a lot faster Sync thing updated and if we refresh the page, you'll still see it's running the old release so There we have it still running the old release It just doesn't do it and I've done it here and done it with the plug-in part so me I'll work on a bug report with that and See what needs to be done, but yeah, it's other than that error I kind of minor but definitely something to think about but I'll be doing some videos on how to manually build the jails How to manually update some of the things and in some future upcoming projects? I have with FreeNAS and IO cage but just there's a lot of documentation on how to do it out there I just haven't done a tutorial on it So you can already read the documentation learn how to do it if you want but tutorials help a lot of people Which I believe is kind of popular with this channel. All right, and thanks And thank you for making it to the end of the video if you like this video Please give it a thumbs up if you'd like to see more content from the channel hit the subscribe button and hit the bell icon If you like YouTube to notify you when new videos come out if you'd like to hire us head over to laurance systems com Fill out our contact page and let us know what we can help you with and what projects you'd like us to work together on If you want to carry on the discussion head over to forums.laurancesystems.com Where we can carry on the discussion about this video other videos or other tech topics in general even suggestions for new videos They're accepted right there on our forums, which are free Also, if you like to help the channel in other ways head over to our affiliate page We have a lot of great tech offers for you and once again, thanks for watching and see you next time


Hey everyone welcome to the latest edition of Nets post game plus Chris Sheeran Frank I solo with you without Kevin Durant and Kyrie Irving Saturday the Nets fall to the Sixers in Philadelphia 124-108 however that score is not indicative of where this game was in the third quarter tied at 79 but that's when it all fell apart for Brooke yeah the last five minutes of that quarter 23 to 7 run by the Sixers Danny Green knocked down a couple of corner three pointers right in front of the Nets bench Harden did not play the fourth quarter and an issue the Nets will have if they end up meeting Philadelphia down in the playoffs will be trying to contain Joe Lmb who's playing like an MVP was outstanding tonight as well as Ben Simmons we have time for that though Frank Joe Lmb did lead all scores with 33 James Harden was the high man for Brooklyn with 26 and some reserves took advantage of extra minutes including big man Norvell Pell who made his Nets debut and blocked three shots in 17 minutes. Steve Nash talked about Pell afterwards but first he was asked by our Michael Grady what caused things to get away from his team in the third quarter. But of everything you know that it's a strong team you know we're a little under man we I thought our our guys got a little tired from the low they're carrying and you know they were able to use their depth and talent to kind of find a little separation but I was proud of the way the guys competed and approached the game. On that point what is your big takeaway you mentioned being proud of the way they played what's the what do you pull away from this game in totality especially when you consider the last 24 hours. Yeah I thought that you know we need to get a little more physical and at times we showed we were we we got in there and cracked people on rebounds double on the floor for loose balls some of the things that we're gonna have to do here to get better but I thought we also cleaned up a little bit of our defensive schemes at times and you know we grew a little bit so you know it was a tough loss we were a little under man but I thought there was a lot to be proud of from our guys. Alex Schiffer the athletic. Hey Steve I got two quick ones for you just early impressions of Norville he was able to get some run and then it looked like the Andre took a shot I'm just wondering if he's alright. I think DJ's okay I'm not positive and Norville was great I mean he's out of shape a little bit for NBA standards you know you can see him out there I was proud of him for for fighting through trying to get his legs and his lungs going he obviously has some instincts blocking shots which is which is I think you know his reputation and why why we looked at him and you know I was proud of the way he he conducted himself and worked hard and like I said he showed that he can block some shots. Brooklyn will return action Tuesday with the road game in Detroit coverage starts on yes at 6 30 with the pregame show presented by Fandle sportsbook for Frank I solo I'm Chris Shearn. Thanks for watching Nets Postgame Plus.


Hi everyone, Joe for Jaspy's casebreaks.com, doing a 2017 Panini Prestige Football Value Pack, random team number three, nine hip-rate football spots being given away. Big thanks to this group right here, and there are the teams right there. Let's do the break first, and then a different dice roll for the giveaway part. Three and a six, nine times. One, two, three, four, five, six, seven, eight, and nine. Yeah, Sean Green was and is still popular in Los Angeles. All right, Michael Danico, three and a six, nine times for the teams, two, three, and nine in final time. And he's also, Sean Green's also from Southern California. I guess they move, I'm looking at his Wikipedia page right now. He was born in Illinois, moved to Jersey when he was one, then to San Jose, and then finally settled down in Tustin, California, when he was 12 years old and then played in high school here. So he's a SoulCal guy, also why he's beloved as a Dodger. All right, so here's no trade windows in these filler breaks. Here's the first half right here, the second half right here. Yeah, Daniel's on, Dilo is on Sean Green's baseball reference, 850 OPS. Yeah, he's pretty good. Yeah, 2001, 49 home runs, 225 RBIs, 20 stolen bases, slash 297, 372, 598. Pretty good. There is the famous, what, four home run game, five home run game? He had against the Brewers, I think. Here's the final printout. Four home runs in a game right, Oliver. That was only sixth in MVP voting? Yeah, I remember he didn't win the MVP that year. Got Theoretic, and got Corey Coleman. Wait, this is 2017, right? So is this like Mahomes rookie card possibilities? I think Mahomes is 2017. But no, I don't think there's, oh right, Barry. That's right. All right, so nothing too crazy there, but let's see who's going to win those hip parade spots. We're giving away nine. So let's gather everybody's names. Once again, for the second and final dice roll, top nine after four, one and three, four times, four down territory, one, two, three, and four. So from 10 on down, sorry Niko, 10 through 32, apologies to everybody there. We'll get them next time. Top nine, you're in the next break, or you're in the hip parade break after another filler. Anyway, it's a different video. Craig, you're happy. So is Simon, Michael, Ryan, Josh, Simon, Josh, Michael, and Niko. You did get it, Niko. Snuck right in there. Name on top after four. Thanks, everybody. I'll see you for the next filler. Then the hip parade breaks.


Welcome to the Knuckleheads of Liberty. In California, the idea for fighting inflation is to throw more money at it. So Gavin Newsom is going to start sending out checks to all of us. And, you know, I guess we're living here, so I suppose we'll be getting those checks all of us.  It's kind of like the whole school debt issue. You know, oh, God, school's getting more expensive. What are we going to do? Oh, we'll throw more government money at it. That'll solve the problem. What do you guys think about this? Well, I did a lot of research preparation for this. Well, as much as I had my time allotted to find out where does this money go when, you know, it's in surplus.  Of course, there is something. And, you know, those checks get cash and they're put into the general fund or the coffers of the California state government. I don't know where that is. I can't find it for the life of me. Maybe Leon knows off the top of his head, but the point I was trying to do to make about this whole thing is, first of all, this is already tax money that they confiscated from us. And so it's free in that respect. I divide money up into two types. I don't know if it's free money, brand newly printed with no, you know, of course, all fiat currency, but I mean is brand new printed money, new stuff that's never been printed before and then boom, it gets spent by the government right into whatever it is. Now, in California, they don't have a printing press. They don't have a central bank, so they can't do that. However, they received $26 billion in federal funds. I don't know if that was just last year or the year before. So you can say that all that money was brand new money. So you can say that they're returning some of that brand new money, or you can say that they're just returning the old money that was already in the possession of the taxpayer before it was taxed away from them and into the state coffers. So in that sense, it's non-inflationary if it's the old money, if it's the new money it is, but it's already been, it's just new money that's already been factored in to the, or its effect has been felt by the inflationary pressures on the economy already. And it's just finally making its way into the hands of the people. So, you know, this is a roundabout way of saying I'm kind of on the fence as to whether this is truly inflationary or whether it's just simply changing the possessor of the money back to its original person that earned it, that taxpayer. What do you think, Leon? Well, to the extent, I see a point, Tim, but to the extent that money is fungible because California do receive, does receive quite a lot of federal funds, and you are right, they did receive a chunk of money. I think it was last year, actually, about that 26 billion you were speaking about, they did receive that from the federal government, which was newly printed. So, if you think about money as being fungible, you can see where the inflationary effect could occur, but you are right, there's the only one saving grace in all of this, that California itself does not have a printing press where they could be printing dollars, right? Thankfully. Yes, we will thank the good Lord for that. But that money could still have a inflationary effect because money is fungible. It doesn't matter that this dollar came from the federal government or that dollar came from the California taxpayer. But this is just socialism at its worst to start with, okay? Because they're taking money away from us, they're working, and well, maybe some of them in the middle class too, and yeah, maybe some of them poor too. And they're just returning it, they're just distributing it to everybody. This is socialism, okay? This is socialism. We don't need, we don't need another stimulus check to suppose, to supposedly, you know, oh my goodness gracious, we're going to go spend it money and all everything is going to be nice and wonderful. This is going to increase demand anyway, increasing demand. It's going to have some inflationary effect from the increase in demand. If demand is increased and there's no change in supply, of course, then it will have some price effects and that price effect will be inflationary. So they're not doing anything for us, okay? They're not doing anything except buying votes. That's what they're doing. Well, so you're saying that if it was for sure exactly that person's tax money that was returned to them, then you're okay with it. But in this case, it seems like, you know, the predominant payers of California state income tax is the rich. So you're saying they're transferring wealth from the rich to all the general population, mainly poor people in, or lower income people, you know. That's what they're doing. That's what you're saying? Yes. If you look at the description of what they're going to do, there are limits on income limits on who could get this so-called stimulus. You're right. In that sense, you're absolutely right. But I still am on the fence about its inflationary aspect. And again, it's only, in my mind, it's only due to whether it's old money or new money. So for example, you take old money, some rich guy earned something.  Okay, that's still old money. Okay, then they turned around and they gave that money to poor people and just say, okay, we're not going to spend it on infrastructure or the schools or any of our pet projects. We're going to give it back to these poor people. So in that sense, you would say that the poor people have a tendency to spend it right into the economy. So in that sense, it would be maybe partially inflationary because it's going from someone that would invest it somewhere else, probably more of a long-term investment like in a stock or such. And somebody that's going to go out and spend it on groceries and gas and things like that. So in that sense, it might have that inflationary push, even though it's all old money. It's kind of a distortion in the economy because what it's doing I think is it shifting it towards the goods that everybody is concerned about going up. And that's literally what they're trying to give them the money to buy more of. The idea is that the price signal is telling us to conserve if it's going up, right? I mean, that's kind of literally what it's trying to do is tell us there's something that's... And so maybe we have to tack on whatever that is we're using instead of, hey, let's shovel some more money into your hands to go buy more of it. So I don't know. But yeah, I agree with you, Tim, though in principle because it's the state and they're not printing money, it's not changing the net amount of money. Maybe the speed of money is changing a little bit for one reason or another. Life, liberty and the pursuit of happiness always and forever.


So, we were discussing the properties of continuous functions. Let us recall that we had proved the following that if you take a function f from x to y, then f is continuous if and only if inverse image of any open set, inverse image of every open set in y is an open set in x and that is equivalent. And this is something quite useful in deciding about the continuity of a function and also proving several properties of the continuous functions. Is it clear to you that from this it follows that if x is a discrete metric space, then every function on it is continuous. Is it clear? Because whatever open set you take here, its inverse image, whatever be the set in x, since in x every set is open, so any function defined on a discrete metric space is always continuous. Now, let us also record a few more things that will follow from this.   And if g of x is not 0 everywhere in x, then f by g is also continuous and here by continuous means continuous on x everywhere. Of course, we can also write a theorem at a particular point. So, if g x is not equal to 0 for every x in x, then f by g is continuous on x. As I said, here I have standard theorem for continuous on x, but you can as well state it for continuity at a point. Instead of say on x, if I change this that suppose both are continuous at some point, let us say point x not in x. Suppose f and g are continuous at a point x not in x, then f plus g is continuous at that point x not, f g is continuous etcetera. And the last thing will be if g x not is not 0, then f by g is continuous at that point x not. Now, as we have seen as far as the proof of any of these things are concerned, it will follow from two things. You just make two cases that to show that function is continuous at all points in x, just take any arbitrary point. Suppose that point is let us say x not, if that is an isolated point, then it is already continuous, there is nothing to be proved. If it is a limit point, then the required conclusion will follow from the corresponding theorem about the limits. For example, we have already proved that limit of f x f plus g of x as x goes to x not that is same as for example, that is for example, we have proved that limit of f x that is f is continuous at x not, f is continuous say if x not is a limit point saying that f is continuous at x not means this. Similarly, g is continuous at x not will be limit of g x as x to x not is g of x not and then by the corresponding theorem of the limits, it will follow that limit of f plus g at x not is same as f x not plus g x not and that will show that f plus g is continuous at that point x not. Since x not was arbitrary on the whole of x and similarly other two things also can be proved. So, the point is conceptually there is nothing new in this, all these assertions will follow from the corresponding assertions about the limits. Then we can also say one more thing here suppose we take let us say k functions let us say f 1, f 2, f k suppose these are k functions from x to r. Suppose each let us I will talk of continuity for the time being let us just say these are real value functions then I can construct what is called a vector value function that is function which goes from x to r k.  Similarly, other way also if you take any function x any function f from x to r k that will give rise to k real value functions. So, just a similar thing as we have seen in the case of sequences now what we want to say is the following if all of these functions are continuous then f is continuous and conversely if f from x to r k is continuous then each of these must be continuous by the way these are sometimes called coordinate functions given by this function f. So, assertion of this f is continuous 1 x of course, here when I say r k I should say what is the metric on r k, but actually it does not matter whichever metric you take this theorem is true, but to just make the just to make the things concrete let us take the metric given by let us say in norms of x 1, but the same more or less the same proof will work for any other metric or any other metric given by any of those L p norms f is continuous on x if and only f j is continuous on x for each j and to see this all that we need to see is the relationship between the absolute value in r and the norm in r k. Since, I have already used this small x for the points in x here let me use something else for the point in r k. So, suppose I take a point let us say a in r k. So, a is a 1, a 2, a k. So, what we know is the following that is norm of a suffix 1 norm of a suffix 1 this is nothing, but sigma mod a j j going from 1 to k and so if so in particular each of this mod a j. So, each of this mod a j is less than or equal to norm of a each of this mod a j is less than or equal to norm of a.  So, f is continuous at some x naught let us say suppose f is continuous at some then let us say we want to put that each of these are continuous at x naught.  So, once you realize this inequalities the proof of this is immediate for example, suppose f is continuous on x let us say f is continuous at x naught for each x and we want to show that each f j is continuous at x naught. Now, what is the meaning of saying that f is continuous at x naught it means given any epsilon given any epsilon you can find a delta such that whenever distance between x and x naught is less than delta distance between f x and f x naught is less than epsilon. That means whenever distance between x and x naught is less than delta this is less than epsilon, but if this is less than epsilon this also should be less than epsilon because f because this quantity is less than or equal to that. So, that shows that f j is continuous and that will hold for each j and what will be the reverse argument? Suppose, we assume that each f j is continuous then again similar thing which we have done that is if each of these let us say if each of these can be made less than let us say epsilon by k then the sum will be less than epsilon. So, we can say if each f j since each f j is continuous we can always find a delta such that whenever distance between x and x naught is less than delta mod f j x minus f j of x naught is less than epsilon by k. This we can do for each k this we can do for sorry this we can do for each j and then if each of this is less than epsilon by k the sum is less than epsilon. So, that will mean that norm of f x minus f x naught is less than epsilon. So, basically the whole thing follows from this and once you realize this you will see that if I take some other norm here instead of let us say norm of f x 1 suppose I have taken norm of f x 2 or non of f x infinity whatever you do this similar argument will work. Now, let us also recall one more thing we have seen that a function is continuous this is this implies that if you are take let us let us let me start from this suppose let me take any other way suppose f from x 2 y is a continuous function. Then we know that it from this it follows that if you take any convergent sequence in x its image in y must be convergent sequence because we have seen that equivalent of two definitions of limit. So, f is continuous means for any let us for any x naught limit of f x as x tends to x naught is f x naught. So, if you take any sequence x n converging to x naught f of x n must converge to f of x naught. So, we can say that f from x 2 y is continuous this is actually if and only if for if x n for every convergent sequence in x let us say sequence x n in x converging to converging to let us say x naught in x f x n converges to f x. So, continuous maps for the property that image of every convergent sequence in x is a convergent sequence in y. Now, let us ask next question what can we say about Cauchy sequences can we say that image of every Cauchy sequence is also a Cauchy sequence. To answer this question best thing is we will look at an example I will take x as the open interval 0 to 1 with the usual matrix and say I will take y as r then define f from x 2 y by f x is equal to 1 by x is that a continuous function that is a continuous function. Suppose I take x n is equal to 1 by n that is a sequence in x of course, it is not a convergent sequence it is not because 0 is not in x it is not a it does not converge to any point in x, but that is not important is it a Cauchy sequence it is it is a Cauchy sequence what is f x n f x n is n f x n is n is that a Cauchy sequence in r it is not right it is not. So, what does it mean in general image of a Cauchy sequence under a continuous function need not be a Cauchy sequence though we know that image of a convergent sequence under a continuous function will always be a continuous function image of a Cauchy sequence need not be a Cauchy sequence. Basically, to take care of this kind of things and of course, for few other things we now discuss a little stronger type of continuity that is called uniform continuity. Now, even before coming to this definition uniform continuity let me also introduce one or two more terminologies more definitions let me remove this for the time. Suppose, f x and y are two matrix spaces and f from x to y is continuous let us say suppose in addition f is also a bisection suppose f is a continuous and bisection. Bisection means it is it is 1 1 and on to and we have known that if a if a function is a bisection we can define what is meant by inverse function. So, then we can define then f inverse is a function from y to x, but f inverse may or may not be continuous if f is continuous and bisection inverse function exists, but at inverse function may or may not be continuous in general unless we put some additional conditions on x y or f etcetera. But, when it is also continuous that map f is called a homeomorphism. So, we will say that f is called a homeomorphism homeomorphism if f from x to y is continuous it is bisection and f inverse from y to x is also continuous. We can give obvious examples of homeomorphisms for example, an entry map going from x to x that is continuous it is inverse is also continuous also we can also construct a non trivial example that is not very difficult. Let me give you as an exercise construct an example of a function which is continuous and bisection, but whose inverse is not continuous. So, let me just give it as an exercise. So, give an example of f from x to y that is continuous bisection, but and f inverse is not continuous that means it is continuous bisection, but not homeomorphism f inverse from y to x is not continuous. Is this clear what is meant by homeomorphism what should happen is that the function f should be continuous inverse should exist. So, it is a bisection and inverse also should be continuous. Now, if you are given two metric spaces we say that those two metric spaces are homeomorphic to each other we say y is homeomorphic to x if there exists a homeomorphism from x to y. So, given we say that y is homeomorphic to x y is homeomorphic to x if there exists a homeomorphism from x to y of course, there is nothing unique about homeomorphism given to metric spaces there may I mean there may not exist any homeomorphism there may exist several homeomorphisms also. Now, this establishes a relation between the set of all metric spaces if you take the class of all metric spaces this says how two metric spaces are related to each other is this is it clear to you that this is an equivalence relation x is homeomorphic to itself that is clear identity map is a homeomorphism if x is homeomorphic to y does it follow that y is homeomorphic to x that is clear if f is a homeomorphism going from x to y f inverse is also a homeomorphism going from y to x what about transitivity suppose we know that x is homeomorphic to y and let us say y is homeomorphic to z will it follow that x is homeomorphic to z obviously, because of what you do is that just take the composition of two homeomorphisms and show that the composition is also homeomorphism and to do that we shall use the theorem that we produce today that composition of two continuous maps is continuous all right. So, this is an equivalence relation and as we have seen earlier any equivalence relation will partition the given class into what are called equivalence classes. So, every metric space in that equivalence class will be homeomorphic to all other metric spaces in that equivalence class all right. Now, let us come to this uniform continuity now here we talk of uniform unlike we can talk of continuity at a point, but we talk of uniform continuity on a set uniform continuity on a of course in particular the set can be a singleton set, but that will be trivial. So, let us just take this case suppose x and y are two metric spaces and as I said yesterday when the metric underlying matrix are not very important for discussion I will not say explicitly that x d is a metric space or y rho is a metric space etcetera whenever that is required we will we will see that for the time being let us let me take this let us say x d and y rho are metric spaces and suppose f is a map which goes from x to y f is a map which goes from x to y and suppose let us say it is continuous suppose x to y is continuous. Now, let us recall the definition of continuity here once again what we have said in the definition is that is that given any epsilon bigger than 0 that is f is continuous means f from x to y is continuous means f is continuous at each x f is this means f is continuous f is continuous at x for every x in x for every x in x f is continuous at x. Now, what is the meaning of this it means that that is for every epsilon bigger than 0 for every epsilon bigger than 0 there exists delta bigger than 0 such that for all y in x distance between y and x less than delta implies distance between f y and f x is less than epsilon. Now, the whole idea of uniform continuity deals with how does this delta depend on epsilon we know that delta may depend on epsilon if epsilon is changed delta is to be changed, but it is not made explicitly clear the definition of continuity that the delta will depend on this x also in general. For example, suppose I take some other point is x 1 or x 2 then it is not it is not necessarily the same delta will work for that x also. So, if it so happens that you can find some delta which works for all elements in some set which works for all elements in some set or in particular for the whole of x then we say that f is uniformly continuous on that set a and if a is equal to x we say f is uniformly continuous on x. So, let us say that let me just get this definition. So, suppose now a is a subset of x f is said to be uniformly continuous on a uniformly continuous on a if for every epsilon bigger than 0 there exists delta bigger than 0 such that this delta will work for all elements in a this delta will work for all elements in such. So, whenever you take any x in a and any y in x right. So, what we say this is important such that for every x in a for every x in a and let us say y in x distance between x and y less than delta this implies distance between f x and f y less than x. Remember once again uniform continuity is always discuss with respect to a set we always say a function is uniformly continuous on a set. Whereas, continuity we can say that function is continuous at a point or on a set both are possible and of course, the special case is when a is equal to x when a is equal to x then f we say f is uniformly continuous on x all right. Of course, a can be a singleton set, but it only means that f is continuous at that point and if a has only one point obviously whatever delta works for that point it works for all points in that set a. So, if a is only one point then any function continuous at that point is also uniformly continuous on a is it also clear that with a small modification of this we can also make this work if a is a finite set. Suppose a is a finite set if there are finite number of points given any epsilon we can find let us say there are let us say there are n points a 1 a 2 a n. So, for each of those a 1 a 2 n you can find delta 1 delta 2 delta n suppose delta 1 works for a 1 delta 2 works for a 2 etcetera delta n works for a n then what you do obviously you take the minimum of all those deltas you take minimum of all those deltas that will work for that will work for every point in that set a. So, what we have essentially set just now is that every function which is continuous it will always be uniformly continuous on a finite set every continuous function will always be uniformly continuous on a on a finite set. So, that is not very important. So, really uniform continuity really matters when we take a as an infinite set and that may or may not happen a function may be continuous, but it may not be uniformly continuous. Let us just see some examples. So, that these ideas will be clear. So, let me start with familiar spaces. So, let us say f from R to R by the way let me again go back to our favorite space discrete metric space we already seen that every function define on a discrete metric space is continuous is it also clear that it is uniformly continuous also. Suppose suppose x suppose x is a discrete metric space then what I can do is that whatever be the epsilon given I can take this delta to be half suppose you take this delta to be half then d x y less than delta can happen only when x is equal to y and in that case distance between f x and f y will be just 0. So, on a discrete metric space not only that every function is continuous, but it is also uniformly continuous. Now, let us take this space and let us look at this function f x is equal to 2 x we know that this is a continuous function the question is whether it is uniformly continuous or not.  So, mod f x minus f y this is nothing, but mod 2 x minus 2 right. So, this is nothing, but 2 times mod x minus y and we want to choose delta in such a way that whenever mod x minus y is less than delta this should be less than epsilon. Now, the choice here is obvious you take just delta is equal to epsilon by 2 delta is equal to epsilon by 2 and this delta will work regardless of what are x and y as soon as mod x minus y is less than epsilon by 2 mod f x minus f y is going to be less than epsilon. So, this is an example of a uniformly continuous function right.  Now, let us see some other function which is not uniformly continuous. Now, before proceeding further let me give you an exercise on your own show that sin x is uniformly continuous on r. Let us take another function g from r to r suppose I take g of x is equal to x square. Now, let us try to do the same thing that is given epsilon bigger than 0 try to find delta satisfying the requirement. So, let epsilon be bigger than 0 then what is g of x minus g of y.  Now, what we want when mod x minus y is less than delta mod x minus y is less than delta this whole thing should be less than epsilon. Now, how do we choose delta like that? Yeah, but y is something that will depend let us say you see what we want is this mod x minus y less than delta this should imply mod x minus y is mod x minus y into mod x plus y this is less than epsilon. So, what we have to do is that let us see that whenever mod x minus y is less than delta what is the maximum possible value of this factor mod x plus y. So, we can say that mod x plus y we can say that this is I can write this as mod x minus y plus 2 y or better still let us say we are talking about continuity at x we can write it as mod y minus x plus 2 x. So, this will be less than or equal to 2 times mod x plus delta 2 times mod x plus delta. So, if I take if I if I choose. So, this will be what I want is delta into 2 times mod x plus delta this should be less than epsilon this should be less than epsilon. Now, we can see the idea the one the way in which one does this is that see we know that if any particular delta works any delta smaller than that will also work any delta smaller than that will also work. So, I can also make another decision that I will choose delta smaller than mod x. Of course, to choose delta smaller than mod x we must require that mod x is not 0 fine we can make that if mod x is equal to 0 that means x is 0 for that we shall think of something else for that we shall think of something else that case we can discuss separately. Suppose, we take the case mod x is bigger than 0 then if I choose delta in such a way that delta is less than mod x then this whole thing will be less than 3 times mod x. So, that means I choose then we have to choose delta in such a way that delta into 3 times mod x is less than epsilon. So, that means delta should be less than epsilon by 3 times mod x that is the requirement and we also want delta is less than mod x. So, we can say that look at minimum of these two numbers minimum of mod x and 1 by 3 times mod x and take delta to see if mod x is bigger than 0 if x is not 0 this number is going to be strictly bigger than 0 choose any delta which is smaller than this that is 0 less than delta less than this that delta will work that delta will work. But it is clear from this discussion that this choice of delta depends on x choice of this delta depends on x if you change the point x the value of delta will change value of delta will change. So, we cannot the same delta will not work for any arbitrary x here. So, this function is not uniformly continuous this function is not uniformly continuous on R. But on the other hand suppose I take some close that boundary interval in R instead of taking the whole of R suppose I take a as simply 0 to 1 suppose I take a as simply 0 to 1 then what you can do is that take the maximum possible take this minimum over all x which belong to that and then take the minimum value of that and then we can find the delta which works for all elements here. So, it is not uniformly continuous on R, but it is uniformly continuous on sets like this it is uniformly continuous on sets like this. So, uniform continuity as I said earlier uniform continuity is a matter on depends on the set on which you are talking about uniform continuity if you change the set if function may be uniformly continuous on some set A it may not be uniformly continuous on some other set B. Now, let us go back to the whole point we started with we said that given an arbitrary continuous function image of a convergent sequence is always convergent, but image of a Cauchy sequence need not be Cauchy. Now, does uniform continuity corrects that view that is can we say that if a function is uniformly continuous then image of a Cauchy sequence is again a Cauchy sequence. Now, before going to that question let me again give you an exercise the same function which we discussed take x is equal to 0 to 1 and y is equal to R and f of x is equal to 1 by x check whether f is uniformly continuous this is an exercise check whether f is uniformly continuous. Now, coming back to that question so suppose f from x to y is uniformly continuous uniformly continuous and x n is equal to is a Cauchy sequence in x x n is a Cauchy sequence in x f x n is a Cauchy sequence in y. So, this shows that under uniformly continuous function image of a Cauchy sequence is a Cauchy sequence anyway how does one show that something is a Cauchy sequence just go by the definition take epsilon bigger than 0 then you have to find some n 0 such that whenever n m and n both are bigger than equal to m 0 distance between f of x n and f of x n should be less than epsilon. So, let epsilon be bigger than 0 then for this epsilon we will have to find n 0 satisfying those requirements, but we know that f is uniformly continuous and x n is a Cauchy sequence. So, these things we should use these things we should use first of all f is uniformly continuous that means for this epsilon there will exist some delta such that whenever distance between the two points here is less than delta distance between their images here is less than epsilon. So, because since f let me now give some matrix also x d and y rho since f is uniformly continuous continuous of course I mean uniformly continuous on x uniformly continuous there exist delta bigger than 0 such that f is uniformly continuous for any two points x and y for any two points x and y in x distance between x and y less than delta this implies distance between f x and f y is less than epsilon. And remember here that it is important to note that this delta does not depend on what is x and y this is important this part is important for every x and y in x this same delta works unlike that function there where delta will depend on the corresponding choice of x. Now we know that x n is a Cauchy sequence. So, what I can say is that for this positive number delta there will exist some n 0 such that whenever n and m is bigger than equal to that n 0 distance between x n and x m is less than delta right. So, since x n is a Cauchy sequence in x since x n is a Cauchy sequence in x there exists n 0 in n such that m and n bigger than or equal to n 0 implies distance between x n and x m is less than delta. Now if distance between x n and x m is less than delta what about distance between f of x n and f of x m that should be less than epsilon because you take x as x n and y as x m here. So, this implies distance between f of x n and f of x m is less than epsilon. So, we have proved that image of a Cauchy sequence under a uniformly continuous function is again a Cauchy sequence which is this is a property which is not true for an arbitrary continuous function. So, uniform continuity is a stronger type of continuity obviously every uniformly continuous function is continuous, but the common is false we have seen the we have seen the examples. Now, let us see some common examples or popular examples of uniformly continuous functions. In fact, some of these types are also given certain particular names one of those is what is called Lipschitz continuous function Lipschitz continuous Lipschitz is again a common name of mathematician perhaps you may have heard of this Lipschitz continuity while discussing differential equations. So, again let us say that take two metric spaces x d and y rho metric spaces and suppose we take a function f from x 2 y function f from x 2 y this is called Lipschitz continuous this is called Lipschitz continuous on x this is called Lipschitz continuous on x if there exist some alpha bigger than 0 if there exist some alpha bigger than 0 such that such that distance between if suppose you take any two points x and y here the distance between f x and f y is that is not equal to this number alpha times distance between x and y. So, what should happen is that for every x and y in x. For every x and y in x distance between so rho f x f y is less than or equal to alpha times distance between x and y. Now, is it obvious that every Lipschitz continuous function is uniformly continuous because you can just take delta as epsilon by alpha suppose you take delta as epsilon by alpha whenever this is less than delta distance between f x and f y will be less than epsilon. So, every Lipschitz continuous function is uniformly continuous of course, it is natural to ask here what about the converse, but we shall not going to that question right now. Now, another special case of this if this constant alpha is strictly less than 1 then this map is called a contraction Lipschitz continuous function is called a contraction if alpha is less than 1. So, if it is called a contraction alpha is less than contraction or contraction map it is called contraction or contraction map. See suppose I had not defined this Lipschitz continuity at all and suppose I want to define contraction directly then what I should have said is that there exist alpha such that 0 less than alpha less than 1 and rho f x f y less than or equal to alpha times d x y for every x y in x that will be the difference of contraction. Here since we already defined what is Lipschitz continuous function contraction is a special case of Lipschitz continuous function and you can also see why it is called contraction. It is called contraction because the distance between the images will be strictly less than the distance between x and y. So, the distance is shortened that is why it is called it is called a contraction and I will just make one more definition and then we will stop with that. This is also something which you come across very often and it is what is known as isometric. Isometry is a map which preserves the distance f is called an isometry if distance between f x and f y is same as distance between x and y. So, f is said to be an isometry if distance between f x and f y is same as distance between x and y for every x y in x. So, all this distance Lipschitz functions, contractions, isometries all of these are examples of uniformly continuous functions and we shall come across the examples like this. It is also it is clear from the definition of an isometry that isometry will be always 1 1 that is clear because suppose f x is equal to f y it will mean that rho f x f y is 0 that will mean the distance between x and y is also 0 that will mean x is equal to y. So, isometry is always 1 1 always 1 1 it may or may not be on to isometry may or may not be on to, but if it is also on to we say that x is isometric to y. We say that x is isometric to y if there is a on to isometry if there is an on to isometry will always be 1 1 by definition if there is an on to isometry means it automatically becomes a bijection then we say that x is isometric to y and this also establishes relation between the two metric spaces just like homomorphism when we say x is homomorphic to y. So, similarly we say x is isometric to y if there is an isometry from x on to y this is also an equivalence relation will stop with that.


This is Shepard with a Jackson Hole shooting experience and I received an email yesterday from a lady that shot with us And she's interested in purchasing a 1911 pistol that is very much like the one that she shot In our multi-gun shooting experience and I thought hey this response that I want to give deserves a little bit more attention than Then just a quick email. So yeah, what don't I you know, I haven't shaved in a few days. I'm on vacation Why don't I make a video? So I decided to do that. So here you go. Hopefully this will help you as well If you're searching for a 1911 so the 1911 that this lady shot was a Springfield Armory 1911 it was a full-sized Some 1911's are made in a altered size the smaller officer model types Compared to the full size. So the one that she shot was a full-sized The model number that Springfield Armory gave it was the a1 now a challenge when you're looking for 1911 pistols is that there are so many options There are a lot of different companies that make them some of them make material changes and some only make cosmetic changes Some have better triggers some have better barrels some have better frames And so there's there's a lot to take into Account as you as you contemplate which one you want. So then you turn to me and I say I don't really know Here's here's how I made my decision. I decided I wanted to compete with a 1911 So I went to a local friend. It was a high-end of 1911 builder a custom builder Herb hazen and I said herb. What do I want and I can't afford one of your nice $3,000 pistols. What what can I do? He suggested he's worked on a lot of different 1911 is doing gunsmithing and he said right now the best one out there Is the Springfield Armory? And so if you get that we can then build it up to be the the good gun for you so I purchased the Springfield Armory a1 and Used it for some years in competition My wife was using it at a class and there was evidently I'm not gonna admit that it was my fault but evidently one of the hand loads I did was a squib and so she had a Catastrophic failure and when she fired the one bullet goes down the barrel and hits the other bullet that was lodged in the barrel that's a squib and It caused the magazine to pop out smoke coming all over the place Fortunately, she wasn't hurt, but it cracked the part of the barrel just a little bit one of the little connectors on the barrel So I took the gun back to herb and said hey What do we need to do and he says, you know, let's put a new barrel in and so he said Ed Brown Would be a good barrel. That's the brand So he put an Ed Brown barrel in and then he did some tuning up of the trigger I'm not a gunsmith and I'm not very tech savvy in that way or engineering savvy gunsmithing savvy But he did some sort of polishing to make the trigger be more sensitive and smoother And so that is the gun that we ended up with oh and I put an aftermarket grip on it They kind of had the little cutouts for the finger or a little molded Cutouts for the finger and I find it to be really comfortable. It has some stippling little bumps on it But it's made of a hard rubbery plastic material So those are kind of the modifications done to my pistol years ago So right now I don't know that Springfield Armory offers that exact pistol and if they do I don't know that That's the exact one you want If you're not concerned so much about, you know, how much it costs I Actually would suggest a an STI. They're just a great brand now when I say this those of you watching this who know guns are gonna say oh you've got to have the Buds of black Hawk or or Nighthawk or Everybody's gonna have a favorite high-end custom 1911 and the truth is Probably all of them are pretty good You're not gonna really go wrong if you're spending more than 1500 or 2000 bucks on a 1911 You'll you'll get something good the things that you care about If you're getting the traditional large frame regular size 1911 you care about the trigger. That's the biggest thing that has a good trigger on it now some companies have tried to make good 1911's that are that are for the gentleman shooter that's only gonna shoot once every couple years and This person wants a higher-end gun But doesn't want to have all the customizing done and Kimber is a good name for this So their guns are gonna cost 20 or 30 percent more than other 1911 makers But they've already done the trigger work and and so on and so forth. I've never met a competitor though that uses Kimber It's it's not really a Hardcore users gun. I'm they're gonna be fine. But if you're planning to shoot a lot I'd step up to a different brand than that as I mentioned I like STI but lots of good brands out there and I'll actually put a link to some of them down in the description of the video Just to give you an idea, but the search is not an easy thing to look for so in this particular situation I would say I don't suggest trying to replicate the The exact gun that I have it was okay, but it was a lower mid-end There's no reason to go buy one of those and then put the ed brown barrel in because the existing barrel is gonna be great It's gonna be just fine She mentioned that the grip felt just wonderful was made for her hand That's how I feel about a lot of the CZ heavy-frame pistols and for her this grip Was just perfect my wife Lynn. She says there's a different gun for every body So for this gals body for her hand this particular grip I had felt great I'll put a link to that down below that is if I can find it I'll find one that's pretty similar when I get back home from vacation and so that'll be included in the description So my long story short suggestion would be to get a Rangemaster version of the STI and get it in 9 millimeter just because it's more comfortable shoot It'll let you shoot more which will make you be a better shooter That's what I would suggest if you do want the full-size 45 ACP That's great. They're they're wonderful, but used it for many years and loved it Then go for any Ed Brown less bear STI is it night hawk or night force something like that I'll put a link as I mentioned or the name of it down below Anything like that. That's a couple grand It's gonna be a great gun the shape and this is a big thing the reason that many of us love 1911's is it the angle of the barrel and the frame the angle of the grip is Just feels good to people who like 1911's it feels good and that angle is gonna be the same on any 1911 model Traditional shaped gun, which is what you're gonna find on gunbroker guns America or wherever you're searching That's what you're gonna find and whatever you find Is gonna be comfortable you get that aftermarket grip that feels good to you Hogue is a popular brand the Hogue grips and that might be what I have on mine as I mentioned I'll put the correct answer down below and then just put that grip on if you have already spent two grand on the gun It's gonna have a good trigger You're probably not gonna have any have to have anything additional done if you do Find a local gunsmith and this lady is from outside of the United States. She's in an even better country She lives in Texas. So in Texas, they're gonna be all kinds of great gunsmiths Talk to a friend talk to somebody at the range and they're gonna mention all you gotta use Bob But Bob is the best ever and he's just you know, he's a master gunsmith. Great Bob's gonna do a very good job Make sure you get a recommendation from an actual shooter preferably a competitive shooter They know more and they care more. They're paying for their own ammunition.  He was assigned a particular Gun and he's not gonna be the expert that the local person that actually competes is gonna be so I would talk to a IDPA USPS a Three gun competitor somebody like that and say hey, who's a who's a good custom gunsmith that can do some work on my trigger so that's my overall advice on selecting the pistol and For all of you that have been out to shoot with us Kind of included in our price is a lifetime of shooting emails to me And I'm not a phone talker kind of person but shoot an email and just say hey, I found these three on Gunbroker, what do you think of them? And I'm again? I'm not a gunperson. I'm a shooting person I'm not that much of a gunsmith Engineering kind of guy, but I've made a lot of mistakes over the years and hopefully I can use those mistakes and kind of take a look and say Oh, you know what? I did make this error. Don't don't reinvent this wheel I just take my advice and go for this one or that one or Can't keep looking or something like that. So hopefully these tips have helped you You know who you are I'm not gonna divulge your name and anyone else watching hopefully these have helped you and Please do be in touch shoot an email to me so to speak and I'll be in touch