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.
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