Directional derivatives of Lipschitz functions

摘要

Letf be a Lipschitz mapping of a separable Banach spaceX to a Banach spaceY. We observe that the set of points at whichf is differentiable in a spanning set of directions but not Gâteaux differentiable isσ-directionally porous. Since Borelσ-directionally porous sets, in addition to being first category sets, are null in Aronszajn’s (or, equivalently, in Gaussian) sense, we obtain an alternative proof of the infinite-dimensional generalisation of Rademacher’s Theorem (due to Aronszajn) on Gâteaux differentiability of Lipschitz mappings. Better understanding ofσ-directionally porous sets leads us to a new version of Rademacher’s theorem in infinite dimensional spaces which we show to be stronger then the one obtained by Aronszajn. A more detailed analysis shows that (a stronger version of) our observation follows from a somewhat technical result showing that the behaviour of the slopes (f(x+t (u+v))−f(x+tv))/t ast → 0+is in some sense independent ofv. In particular, this implies that in the case of Lipschitz real valued functions the upper one-sided derivatives coincide with the derivatives defined by Michel and Penot, except for points of aσ-directionally porous set. This has a number of interesting consequences for upper and lower directional derivatives. For example, for allx ∈ X, except those which belong to aσ-directionally porous set, the functionv → $bar f$ (x, υ) (the upper right derivative off atx in the directionv) is convex.