I've recently learned that the Cantor function is Hölder continuous with parameter $\alpha$ where $\alpha$ is the Hausdorff dimension of the Cantor set.
My intuition is that this fact stems from the fact that the Cantor function is essentially the "CDF corresponding to random choice from the Cantor set", I've been wondering if this intuition can be generalized as follows:
Let there be $0<D<1$ and let $C\subset\mathbb{R}$ be a set of Hausdorff dimension $D$ such that $\mathcal{H}_{D}(C)=1$. Define a function: $$f(x)=\mathcal{H}_{D}((-\infty,x)\cap C)$$ is $f$ necessarily Hölder continuous with parameter $D$?