Rademacher's theorem

Rademacher's theorem In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of Rn and f: U → Rm is Lipschitz continuous, then f is differentiable almost everywhere in U; that is, the points in U at which f is not differentiable form a set of Lebesgue measure zero.

Generalizations There is a version of Rademacher's theorem that holds for Lipschitz functions from a Euclidean space into an arbitrary metric space in terms of metric differentials instead of the usual derivative.

See also Alexandrov theorem Pansu derivative References Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, Berlin–Heidelberg–New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325, Zbl 0176.00801. (Rademacher's theorem is Theorem 3.1.6.) Heinonen, Juha (2004). "Lectures on Lipschitz Analysis" (PDF). Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.) This mathematical analysis–related article is a stub. You can help Wikipedia by expanding it.

Categories: Lipschitz mapsTheorems in measure theoryMathematical analysis stubs

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