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 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.)