Condition for continuous derivative

Question

Let $f: [0,1]\to\mathbb{R}$ be continuous and differentiable on $(0,1)$. Is it true that if $m \in (0,1)$ is a local maximum then $f'$ is continuous in some neighborhood of $m$? Intuitively this should looks true, but my attempts at a proof have not gotten me very far. If it false, can you provide a counterexample?

Answer 1

It is not true. Take $f(x)=\left(x-\frac12\right)^2\left(\sin\left(\frac1{x-\frac12}\right)-2\right)$ if $x\neq\frac12$ (and $f\left(\frac12\right)=0$). Then $f$ has a local maximum at $\frac12$, but $f'$ is discontinuous there.