# On differentiability relative to a n-rectifiable subset of $\mathbb{R^N}$

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.