A Lipschitz function which is $C^{\infty}$ on a.e. line need not be generically differentiable
Tom 131 / 2013
Colloquium Mathematicum 131 (2013), 29-39
MSC: Primary 46G05; Secondary 26B05.
DOI: 10.4064/cm131-1-3
Streszczenie
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).
