Derivative of differentiable function need not be continuous

Statement

It is possible to have a function $f$ defined for real numbers such that $f$ is a differentiable function everywhere on its domain but the derivative $f'$ is not a continuous function.

Equivalently, a differentiable function on the real numbers need not be a continuously differentiable function.

Proof

Example with an isolated discontinuity

Consider the function:

$f(x):=\left\lbrace {\begin{array}{rl}x^{2}\sin(1/x),&x\neq 0\\0,&x=0\\\end{array}}\right.$

Then, we have:

$f'(x)=\left\lbrace {\begin{array}{rl}2x\sin(1/x)-\cos(1/x)&x\neq 0\\0,&x=0\\\end{array}}\right.$

In particular, we note that $f'(0)=0$ but $\lim _{x\to 0}f'(x)$ does not exist. Thus, $f'$ is not a continuous function at 0.

For details, see square times sine of reciprocal function#First derivative.