On Gâteaux Differentiability of Pointwise Lipschitz Mappings

Duda, Jakub

Canadian mathematical bulletin, Tome 51 (2008) no. 2, pp. 205-216

Résumé

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.

Détail

DOI : 10.4153/CMB-2008-022-6

Mots-clés : 46G05, 46T20, Gâteaux differentiable function, Radon-Nikodým property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions