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 → R^m

is Lipschitz continuous, then f is Fréchet-differentiable almost everywhere in U (i.e. 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.

References

Juha Heinonen, Lectures on Lipschitz Analysis, Lectures at the 14th Jyväskylä Summer School in August 2004. (Rademacher's theorem with a proof is on page 18 and further.)

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