Typical Lipschitz mappings are typically non-differentiable

TL;DR

This paper demonstrates that most Lipschitz functions between Banach spaces are non-differentiable at most points, revealing a fundamental property of such mappings even in Euclidean spaces.

Contribution

It establishes that typical Lipschitz mappings are almost everywhere non-differentiable, a novel and extreme result in the theory of Lipschitz functions.

Findings

Most Lipschitz mappings are non-differentiable at typical points.

The result holds for mappings between any Banach spaces.

Even in Euclidean spaces, typical Lipschitz functions are non-differentiable almost everywhere.

Abstract

We prove that a typical Lipschitz mapping between any two Banach spaces is non-differentiable at typical points of any given subset of its domain in the most extreme form. This is a new result even for Lipschitz mappings between Euclidean spaces.