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# On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Published:2008-06-01
Printed: Jun 2008
• Jakub Duda
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## Abstract

We prove that for every function $f\from X\to Y$, where $X$ is a separable Banach space and $Y$ is a Banach space with RNP, there exists a set $A\in\tilde\mcA$ such that $f$ is G\^ateaux differentiable at all $x\in S(f)\setminus A$, where $S(f)$ 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 $\tilde\mcC$; this improves a result due to Borwein and Wang. Another corollary is that if $X$ is Asplund, $f\from X\to\R$ cone monotone, $g\from X\to\R$ continuous convex, then there exists a point in $X$, where $f$ is Hadamard differentiable and $g$ is Fr\'echet differentiable.
 Keywords: Gâteaux differentiable function, Radon-Nikodým property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions