When is the derivative of a function continuous?

Question

I know that there exists functions where the derivatives are discontinuous, but most of the examples look sort of artificial. For example: the function $ f(x) = x^2 \sin (1/x)$ when x is non zero and taking value zero when x is zero. Here the original function itself has an undefined point. So I am a bit confused as to whether seemingly ordinary functions can have discontinuous derivatives.

Hence I am looking for any sort of lemmas or theorems which will give hints to as when a function will have a continuous derivative.

Written as $f(x)=\begin{cases}x^2\sin(1/x)&x\ne 0\0&x=0\end{cases}$, the original function does not have undefined points ... — Hagen von Eitzen, Sep 02 '19 at 11:00
This maybe relative-Looks like it is a dominant example: :https://math.stackexchange.com/questions/292275/discontinuous-derivative — NoChance, Sep 02 '19 at 11:02
Sum, differences, products, quotients (where defined), compositions of functions with continuous derivatives are again functions with continuous derivatives. — Hagen von Eitzen, Sep 02 '19 at 11:02
Note also that if $f$ is differentiable on $(a,b)$ and $f’$ has a limit at $b$, then $f$ is differentiable at $b$ and $f’$ is continuous at $b$. — Aphelli, Sep 02 '19 at 12:23
If the (directional) derivative does not exist at a point, it has no limit at this point. Say that $f(x)$ is smooth on $x > 0$ but not differentiable at $0$: then $f’(x)$ has no likit when $x \rightarrow 0$. — Aphelli, Sep 02 '19 at 12:43
I picked the dupe target on the basis of this answer, which I think addresses this question. This answer to another question may also be relevant. — Xander Henderson, Sep 02 '19 at 12:46
A derivative always satisfies the intermediate value theorem, whether it is continuous or not. So if it's discontinuous it can't have a simple jump discontinuity, it has to fail to have a limit entirely. — Matt Samuel, Sep 02 '19 at 20:10
@MattSamuel thankyou for that comment. I was looking for such theorems. — N.S.JOHN, Sep 03 '19 at 04:11

When is the derivative of a function continuous?

0 Answers0