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:

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

Then, we have:

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

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

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

The video below covers this example.