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

Luděk Zajíček. "A Lipschitz function which is $C^{∞}$ on a.e. line need not be generically differentiable." Colloquium Mathematicae 131.1 (2013): 29-39. <http://eudml.org/doc/283567>.

@article{LuděkZajíček2013,
abstract = {We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^\{∞\}$ 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).},
author = {Luděk Zajíček},
journal = {Colloquium Mathematicae},
keywords = {Gâteaux differentiability; Lipschitz function},
language = {eng},
number = {1},
pages = {29-39},
title = {A Lipschitz function which is $C^\{∞\}$ on a.e. line need not be generically differentiable},
url = {http://eudml.org/doc/283567},
volume = {131},
year = {2013},
}

TY - JOUR
AU - Luděk Zajíček
TI - A Lipschitz function which is $C^{∞}$ on a.e. line need not be generically differentiable
JO - Colloquium Mathematicae
PY - 2013
VL - 131
IS - 1
SP - 29
EP - 39
AB - We construct a Lipschitz function f on X = ℝ ² such that, for each 0 ≠ v ∈ X, the function f is $C^{∞}$ 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).
LA - eng
KW - Gâteaux differentiability; Lipschitz function
UR - http://eudml.org/doc/283567
ER -