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

• Autorzy: Luděk Zajíček
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 26B05: Continuity and differentiation questions
• 46G05: Derivatives

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2013
• Tom: 131
• Numer: 1
• Strony: 29-39

Daty

wydano

2013

Twórcy

autor

Luděk Zajíček

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

Identyfikatory

DOI

10.4064/cm131-1-3