Hausdorff measure of the graph

Question

Is there any example of a real valued function on the real line whose domain has Lebesgue measure zero but the graph (in the plane) has positive one dimensional Hausdorff measure?

Of course if such an example exists then it must be not lipschitzian.

Answer 1

Of course.
Restrict Cantor's staircase to the Cantor set.

[Here is the graph of Cantor's staircase. Now remove the (open) horizontal line-segments.]

The domain consists of the Cantor set set $C$, that is reals of the form $$ x = \sum_{k=1}^\infty 2\epsilon_k 3^{-k} $$ where $(\epsilon_k) \in \{0,1\}^{\mathbb N}$. That is, $x \in [0,1]$ has an expansion in base $3$ using only the digits $0$ and $2$. This set $C$ has Lebesgue measure zero.

The image of such a point $x$ is $$ f(x) = \sum_{k=1}^\infty \epsilon_k 2^{-k} $$ The range of the function is $[0,1]$, which has Hausdorff dimension $1$, so the graph of $f$ has Hausdorff dimension at least $1$.