On Gâteaux Differentiability of Pointwise Lipschitz Mappings
Printed: Jun 2008
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.
Gâteaux differentiable function, Radon-Nikodým property, differentiability of Lipschitz functions, pointwise-Lipschitz functions, cone mononotone functions
46G05 - Derivatives [See also 46T20, 58C20, 58C25]
46T20 - Continuous and differentiable maps [See also 46G05]