I'm looking for an elementary example of a function $f$ satisfying
- $f$ is continuous on $[0,1]$, continuously differentiable on $(0,1)$,
- The derivative $f'$ is bounded on $(0,1)$,
- However, the function $f$ is not differentiable at endpoint $x=0$ and $x=1$.
I first tried something like $f'(x)=\sin\frac{1}{x}$, but to my surprise, it turned out that this cannot work since then $f$ becomes differentiable at $x=0$: See this MSE answer for more detail.
I vaguely remember finding a nice example here in MSE using logarithm, but I can't find it as of now. If such a function exists, it would make an interesting example for the second part of the fundamental theorem of calculus.
