Suppose *f* is a real-valued function of one variable, and suppose *f* is of differentiability class C1. My question is, if $\Gamma$ is the graph of *f*, then must $\dim\_H(\Gamma)=1$? If anyone knows of a published proof of the answer, I'd appreciate a reference greatly.



Do all graphs of C1 functions have Hausdorff dimension 1?
Let $f \colon I \to \mathbb{R}$. Since $f$ is $C^1$, the graph $\Gamma\_f$ is locally bilipschitz to $I$, via the projection. It follows that Hausdorff dimension is the same as that of $I$ (being defined in terms of the metric space structure only), so it is $1$.


Disclaimer: I haven't seen these topics for quite a while, so I may have said something stupid.


