Let $S$ be a $n$-rectifiable subset of $\mathbb{R}^N$ , we define the differentiability of a funtion $f:S \to \mathbb{R}$ at a point $x_0$ in $S$ as in Federer's book, where he called differentiable relative to $S$ at $x_0$.

Are there any known condition to ensure that f is differentiable relative to S at $H^n$-a.e. points in $S$?

Here we suppose that S is equipped with a metric $d$ such that it is Ahlfors $n$-regular in Hausdorff measure, meaning that the Hausdorff $n$-measure of balls with radius $r$ in $S$ is comparable to $r^n$. That is, We may view $S$ itself as an $n$-regular $n$-rectifiable metric measure space.