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In mathematical analysis, Rademacher's theorem, named after Hans Rademacher, states the following: If U is an open subset of and f: URm 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.

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In mathematical analysis, a metric differential is a generalization of a derivative for a Lipschitz continuous function defined on a Euclidean space and taking values in an arbitrary metric space. With this definition of a derivative, one can generalize Rademacher's theorem to metric space-valued Lipschitz functions.

• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, 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.)