I am trying to prove this result:
If $f$ is a non-constant function defined on $[a, b]$ such that $f$ is differentiable on $[a, b]$ with a bounded derivative and $f'(x) = 0$ for all $x$ in a dense subset of $[a, b]$ then $f'$ is not Riemann integrable on $[a, b]$.
The conclusion deals with $f'$ and I don't seem to have enough details on $f'$ except that it is bounded and vanishes in a dense subset of $[a, b]$ (meaning every open sub-interval of $[a, b]$ contains points where derivative $f'$ vanishes). Perhaps I am missing some implication of vanishing of $f'$ on a dense set.
Any hints or a solution based on elementary techniques (i.e. not involving measure theory) will be highly appreciated.