Gâteaux differentiability of Lipschitz functions via directional derivatives.

Abstract

Let \(X\) be a separable Banach space, \(Y\) a Banach space and \(f:X \to Y\) a Lipschitz function. We show that the set of all G\^ ateaux non-differentiability points at which \(f\) has all one-sided or two-sided directional derivatives can be covered by (special subsets of) Lipschitz surfaces of codimension 1 or codimension 2, respectively. Further results indicate that these results are close to the best possible ones. Our results are new also for Lipschitz functions \(\R^n \to \R\); for these functions G\^ ateaux differentiability is the classical (total) differentiability.