Directional Derivatives of Lipschitz Functions

Let f be a Lipschitz mapping of a separable Banach space X to a Banach space Y. We observe that the set of points at which f is diierentiable in a spanning set of directions but not G^ ateaux diierentiable is-directionally porous. Since Borel-directionally porous sets, in addition to being rst category sets, are null in Aronszajn's (or, equivalently, in Gaussian) sense, we obtain an alternative proof of the innnite-dimensional generalisa-tion of Rademacher's Theorem (due to Aronszajn) on G^ ateaux diierentiability of Lipschitz mappings. Better understanding of-directionally porous sets leads us to a new version of Rademacher's theorem in innnite 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 as t ! 0+ is in some sense independent of v. In particular, this implies that in the case of Lipschitz real valued functions the upper one-sided derivatives coincide with the derivatives deened by Michel and Penot, except for points of a-directionally porous set. This has a number of interesting consequences for upper and lower directional derivatives. For example, for all x 2 X, except those which belong to a-directionally porous set, the function v ! f(x; v) (the upper right derivative of f at x in the direction v) is convex. 1. Introduction The simple fact that some exceptional sets which arise naturally in the study of (directional) diierentiability of Lipschitz functions on separable Banach spaces are-porous had been pointed out by the authors in several lectures, but we felt that the corollaries were not strong enough to warrant a publication.