We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is 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 dimX > 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).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-3,
author = {Lud\v ek Zaj\'\i \v cek},
title = {A Lipschitz function which is $C^{$\infty$}$ on a.e. line need not be generically differentiable},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {29-39},
zbl = {1322.46027},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-3}
}
Luděk Zajíček. A Lipschitz function which is $C^{∞}$ on a.e. line need not be generically differentiable. Colloquium Mathematicae, Tome 131 (2013) pp. 29-39. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm131-1-3/