Full non-differentiability sets of typical Lipschitz functions

TL;DR

This paper characterizes the sets where typical Lipschitz functions lack directional derivatives, showing they are precisely those contained in countable unions of closed purely unrectifiable sets.

Contribution

It provides a complete characterization of the non-differentiability sets for typical Lipschitz functions in terms of geometric measure theory.

Findings

Typical Lipschitz functions have no directional derivative on certain Borel sets.

Such sets are exactly those contained in countable unions of closed purely unrectifiable sets.

The result links differentiability properties to geometric measure-theoretic structures.

Abstract

In this paper we prove that the typical Lipschitz function has no directional derivative at any point of a Borel set $E$ if and only if $E$ is contained in a countable union of closed purely unrectifiable sets.