$f(x)$ is differentiable at $x = a,$ but $f(x)$ doesn't have a second derivative at $x = a$

Question

Does anyone have an example of a function $f(x)$ which is differentiable at the point $x = a,$ but doesn't have a second derivative at $x = a$?

Answer 1

Consider
$$f(x)= \begin{cases} 0& \text{for } x< 0\\ x^2 & \text{for } x\geq0 \end{cases}$$

then $f'(0)=0$ but $f''(0)$ does not exist.

Answer 2

The primitive(s) of $x\longmapsto |x|$. Namely, $ x\longmapsto \frac12 x|x|+C$.