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On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Published online by Cambridge University Press:  20 November 2018

Jakub Duda
Jakub Duda*
Affiliation:
Department of Mathematics, Weizmann Institute of Science, Rehovot 76100, Israel, e-mail: duda@karlin.mff.cuni.cz
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Abstract

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We prove that for every function $f\,:\,X\,\to \,Y$ , where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\,\in \,\overset{\sim }{\mathop{\mathcal{A}}}\,$ such that $f$ is Gâteaux differentiable at all $x\,\in \,S\left( f \right)\backslash A$, where $S\left( f \right)$ is the set of points where $f$ is pointwise-Lipschitz. This improves a result of Bongiorno. As a corollary, we obtain that every $K$-monotone function on a separable Banach space is Hadamard differentiable outside of a set belonging to $\tilde{C}\,;$ this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\,:\,X\,\to \,\mathbb{R}$ cone monotone, $g\,:\,X\,\to \,\mathbb{R}$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fréchet differentiable.

Keywords

46G0546T20Gâteaux differentiable functionRadon-Nikodým propertydifferentiability of Lipschitz functionspointwise-Lipschitz functionscone mononotone functions
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 2 , 01 June 2008 , pp. 205 - 216
Copyright
Copyright © Canadian Mathematical Society 2008

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