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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.

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

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