We have $f,g: \mathbb{R} \rightarrow \mathbb{R}$ and $g(f(x))=f'(x)$. $f$ is differentiable in all points. Prove that $f$ is monotonic.
It is easy to prove that $f$ is monotonic when $f'$ is continuous, because otherwise we could easily find $a,b$ such that $f(a)=f(b)$ with differing derivatives (because in one point $f$ is increasing and in the other one it is decreasing).
I suspect similar reasoning works when we do not assume that $f'$ is continuous, because there should be a way to guarantee that there are continuous intervals of $f'$, and we could use this to work out solution. However, I couldn't do this in any strict way.