Partial answer. By Frostman's lemma, if $E$ is a compact set of positive $\alpha$-Hausdorff content, then there exists a probability Borel measure $\mu$ supported in $E$ such that $\mu(I) \leq c |I|^\alpha $ for every interval $I$ and $c$ is a constant. Then the distribution function of the measure $\mu$ must be $\alpha$-Holder continuous and $f(E)=[0,1]$.
an_ordinary_mathematician
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