A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable

Luděk Zajíček¹

¹ Charles University Faculty of Mathematics and Physics Sokolovská 83 186 75 Praha, Czech Republic

Colloquium Mathematicum, Tome 131 (2013) no. 1, pp. 29-39.

Résumé

We construct a Lipschitz function $f$ on $X= {\mathbb R}^2$ such that, for each ${0 \not =v \in X}$, the function $f$ is $C^{\infty }$ smooth on a.e. line parallel to $v$ and $f$ is Gâteaux non-differentiable at all points of $X$ except a first category set. Consequently, the same holds if $X$ (with $\mathop{\rm dim}X >1$) 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).

Détail

DOI : 10.4064/cm131-1-3

Keywords: construct lipschitz function mathbb each function infty smooth line parallel teaux non differentiable points except first category set consequently holds mathop dim arbitrary banach space has usual measure sense example gives answer natural question concerning authors recent study linearly essentially smooth functions which generalize essentially smooth functions borwein moors