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What is the difference between continuous derivative and derivative? According to my teachers solution to the assignment,it seems there exits difference between continuous derivative and derivative. However, aunt Google does not tell me what I want.

Edit: Here is a example. $$f(x) = \begin{cases} \frac{1-cos2x}{x} & \text{otherwise} \\ k & \text{if x=0} \end{cases}$$

Does k is continuous but not continuous derivative at $0$?


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This is a little hard to answer without more context - my guess would be that by "continuous derivative" they mean the usual derivative, but are remarking that (for a particular function) this derivative is continuous. –  Matthew Pressland Oct 25 '12 at 13:49
@MattPressland I have posted a example:) –  John Hass Oct 29 '12 at 16:36

2 Answers 2

up vote 14 down vote accepted

The derivative of a function (if it exists) is just another function. Saying that a function is differentiable just means that the derivative exists, while saying that a function has a continuous derivative means that it is differentiable, and its derivative is a continuous function.

To show that a differentiable function need not have a continuous derivative, consider the function $f$ defined by

$$ f(x)= \begin{cases} x^{2}\sin(1/x) & \text{if } x\neq 0\\ 0 & \text{otherwise. } \end{cases}$$

See the following paper if you want to see a full discussion of why this is a satisfactory example:

"A Discontinuous Derivative" (2006) - by Louis A. Talman, Mathematical State College of Denver http://rowdy.msudenver.edu/~talmanl/PDFs/APCalculus/DiscontDeriv.pdf

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Further, in some sense, all derivative which are not continuous must have this kind of crazy oscillation behavior. This is because derivatives are Darboux function, that is, they satisfy the intermediate value theorem whether or not they are continuous. See en.wikipedia.org/wiki/Darboux%27s_theorem_%28analysis%29 –  Jason DeVito Oct 25 '12 at 14:09

A function needs to be continuous in order to be differentiable. However the derivative is just another function that might or might not itself be continuous, ergo differentiable.

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