Directional upper derivatives and the Chain Rule formula for locally Lipschitz functions on Banach spaces

Research output: Contribution to journalArticle

Authors

Colleges, School and Institutes

External organisations

  • Warwick Mathematics Institute, University of Warwick

Abstract

Motivated by an attempt to find a general chain rule formula for differentiating the composition f ◦ g of Lipschitz functions f and g that wouldbe as close as possible to the standard formula (f ◦ g)'(x) = f'(g(x)) ◦ g'(x), we show that this formula holds without any artificial assumptions provided derivatives are replaced by complete derivative assignments. The idea behind these assignments is that the derivative of f at y is understood as defined only in the direction of a suitable “tangent space” U(f, y) (and so it exists at every point), but these tangent spaces are chosen in such a way that for any g they
contain the range of g'(x) for almost every x. Showing the existence of such assignments leads us to a detailed study of derived sets and the ways in which they describe pointwise behavior of Lipschitz functions.

Details

Original languageEnglish
Pages (from-to)4685-4730
Number of pages46
JournalTransactions of the American Mathematical Society
Volume368
Issue number7
Early online date18 Nov 2015
Publication statusPublished - 2016

Keywords

  • Lipschitz functions, Chain Rule, Derivatives, Derived sets

ASJC Scopus subject areas