If a function $ f : \mathbf{R} \to \mathbf{R} $ is $\mathscr{C}^{0,\alpha}$ for every $ 0 < \alpha < 1 $ then its graph has Hausdorff dimension $1$.

I would like to see an example of such a function with a NON $\sigma$ finite graph (with respect to $\mathscr{H}^{1}$).