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.)