Rademacher's theorem

The set of Rademacher, named after the German mathematician Hans Rademacher, is a set of analysis on Lipschitz continuous functions.

Statement

Be natural numbers, an open subset of a Euclidean space, and finally a Lipschitz continuous function. Then almost everywhere (totally ) differentiable.

That is, the set of all points where it is not differentiable, is a Lebesgue null set.

Generalization

There is a generalization of functions, where now denote an arbitrary metric space.

First, however, it is not clear how the above theorem to this case can be transferred, as a metric space carries not a priori a linear structure.

Summing as a function between normed spaces and places the Fréchet differentiability basis, then the sentence is even wrong:

The German mathematician Bernd Kirchheim but now can generalize the theorem of Rademacher in another sense: