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

Colloquium Mathematicae (2013)

- Volume: 131, Issue: 1, page 29-39
- ISSN: 0010-1354

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topLuděk 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 -

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