A counter-example to differential function but not twice differential

Question

Find a function $f$ that is differentiable, but not twice differentiable and which does not belong to the following type: $$f(x) = \begin{cases} x^\alpha \sin(x^{\beta}) & x \neq 0 \\ 0 & x=0.\end{cases}$$ Please give me a hint.

Answer 1

Consider $F$ an antiderivative of $x\to |x|$

Is $F$ twice differentiable?

Answer 2

$x\mapsto x|x|$ should be a good candidate.

Answer 3

Take the anti-derivative of the Weierstrass function which is continuous everywhere and differentiable nowhere.