Does there exist a real valued function on $[0, 1]$ that is differentiable everywhere, but for every $\alpha > 0$ is nowhere locally $\alpha$-Hölder continuous? That is, it is not $\alpha$-Hölder on any open subinterval.

I hope I am not overlooking something elementary, but to my utmost surprise I actually think this might be true…

The answer by user479223 [here](https://mathoverflow.net/questions/460018/can-a-nowhere-locally-h%C3%B6lder-function-be-differentiable-almost-everywhere?rq=1) shows that such a function can be differentiable almost everywhere, in fact even increasing and continuous.