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Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)$
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Joram Lindenstrauss, David Preiss, and Jaroslav Tier

Print publication date: 2012

Print ISBN-13: 9780691153551

Published to Princeton Scholarship Online: October 2017

DOI: 10.23943/princeton/9780691153551.001.0001

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Fr échet Differentiability of Real-Valued Functions

Fr échet Differentiability of Real-Valued Functions

(p.222) Chapter Twelve Fr échet Differentiability of Real-Valued Functions
Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces (AM-179)

Joram Lindenstrauss

David Preiss

Tiˇser Jaroslav

Princeton University Press

This chapter shows that cone-monotone functions on Asplund spaces have points of Fréchet differentiability and that the appropriate version of the mean value estimates holds. It also proves that the corresponding point of Fréchet differentiability may be found outside any given σ‎-porous set. This new result considerably strengthens known Fréchet differentiability results for real-valued Lipschitz functions on such spaces. The avoidance of σ‎-porous sets is new even in the Lipschitz case. The chapter first discusses the use of variational principles to prove Fréchet differentiability before analyzing a one-dimensional mean value problem in relation to Lipschitz functions. It shows that results on existence of points of Fréchet differentiability may be generalized to derivatives other than the Fréchet derivative.

Keywords:   cone-monotone function, Asplund space, Fréchet differentiability, mean value estimate, Lipschitz function, σ‎-porous sets, variational principles, Fréchet derivative

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