H\"older functions with nonporous graphs

For any $\alpha \in (0,1)$ there is a Hölder continuous function $h:\mathbb{R} \rightarrow \mathbb{R}$ with nonporous graph $Gr(h) \subset \mathbb{R}^2$ which is thin for doubling measures.

Construction of these functions requires analysis of a known example with nonporous graph and bounded variation on the unit interval.