Let $f:[0,1]\to[0,1]$ be a devil's staircase in the usual sense. (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where $f'$ vanishes to have Hausdorff dimension zero.
Question. Is it true that $f$ is not Hölder continuous?
(This looks plausible, since $f$ has `very little room' where it can grow so it has to grow very fast - at least, at some points.)