A LIPSCHITZ FUNCTION WHICH IS C-infinity ON AE LINE NEED NOT BE GENERICALLY DIFFERENTIABLE

Abstract

We construct a Lipschitz function f on X = R-2 such that, for each 0 not equal nu is an element of X, the function f is C-infinity smooth on a.e. line parallel to v and f is Gateaux non-differentiable at all points of X except a first category set. Consequently, the same holds if X (with dim X > 1) is an arbitrary Banach space and "a.e." has any usual "measure sense".

This example gives an answer to a natural question concerning the author's recent study of linearly essentially smooth functions (which generalize essentially smooth functions of Borwein and Moors).

Keywords

• Gateaux differentiability
• essentially smooth functions
• Lipschitz function
• functions C-infinity on a.e. line.