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

Edit: Here is a example. $$f(x) = \begin{cases} k & \text{if }x=0 \\ \frac{1-\cos(2x)}{x} & \text{otherwise} \end{cases}$$

Is $f$ continuous but not having continuous derivative at $0$?

Thanks:)

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    $\begingroup$ 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. $\endgroup$
    – mdp
    Commented Oct 25, 2012 at 13:49
  • $\begingroup$ @MattPressland I have posted a example:) $\endgroup$ Commented Oct 29, 2012 at 16:36

2 Answers 2

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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

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    $\begingroup$ 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 $\endgroup$ Commented Oct 25, 2012 at 14:09
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    $\begingroup$ Thanks for the wonderful paper $\endgroup$ Commented Apr 29, 2022 at 9:47
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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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