How to give an example of a $f$ differentiable in a deleted neighborhood of $x_0$ such that $\lim_{x\to x_0}f^\prime(x)$ doesn't exist?

Question

How would I give a simple example of a function $f$ differentiable in a deleted neighborhood of $x_0$ such that $\lim_{x\to x_0}f^\prime(x)$ does not exist? I cannot seem to think of an example.

A delete neighborhood is an open interval about $x_0$ which does not contain $x_0$. So, $(x_0-\delta,x_0+\delta)-\{x_0\}$ for some $\delta>0$.

How would something be differentiable in a deleted neighborhood if at the point of the derivative, the limit does not exist. Presumably, the derivative ends up looking something like $\lim_{x\to x_0} \dfrac{1}{x}$, if it does not exist.

Answer 1

Classic example: $$\sqrt[3]{(x-x_0)^2}$$

Answer 2

Take $f(x) = x \sin (1/x)$ near $0$

Answer 3

You may try $f(x)=x^2\cos(1/x)$, so that $f'(x)=2x\cos(1/x)-\sin(1/x)$ has a point of discontinuity at $x=0$.

Answer 4

Does $f(x)=x^\frac 12 $ count?
$f'(x)=\frac 1{2x^\frac 12}$ which is discontinuous at $x=0$

Answer 5

$$\ln'(x) = \dfrac{1}{x}$$

If you are looking for that exact derivative.