Differentiability and Directional Differentiability in Euclidean Spaces

Abstract

Existence of a derivative implies existence of all directional derivatives, but the converse is not necessarily true. We describe some situations in which one can pass from directional derivatives to differentiability. One situation is when we assume a directional derivative is maximal. This can be applied to obtain differentiability in small sets and study converses to Rademacher’s theorem. Another situation is when we discard a small sigma-porous set. This can be applied to give an alternative proof of Rademacher’s theorem.

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