Differentiability of Lipschitz functions in Lebesgue null sets

Abstract

We show that if $$n>1$$ then there exists a Lebesgue null set in $${\mathbb {R}}^{n}$$ containing a point of differentiability of each Lipschitz function $$f:{\mathbb {R}}^{n} \rightarrow {\mathbb {R}}^{n-1}$$ ; in combination with the work of others, this completes the investigation of when the classical Rademacher theorem admits a converse. Avoidance of $$\sigma $$ -porous sets, arising as irregular points of Lipschitz functions, plays a key role in the proof.