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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.

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I found some results on this topic, except Federer's book "geometric measure theory",

  1. Ambrosio, Luigi; Kirchheim, Bernd Rectifiable sets in metric and Banach spaces. Math. Ann. 318 (2000), no. 3, 527–555. (DOI: 10.1007/s002080000122, preprint, Internet Archive)

  2. Keith, Stephen Measurable differentiable structures and the Poincaré inequality. Indiana Univ. Math. J. 53 (2004), no. 4, 1127–1150 (jstor).

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