On the linear Denjoy property of two-variable continuous functions

• Autorzy: Miklós Laczkovich , Ákos K. Matszangosz
• Języki publikacji: EN

Abstrakty

EN

The classical Denjoy-Young-Saks theorem gives a relation, here termed the Denjoy property, between the Dini derivatives of an arbitrary one-variable function that holds almost everywhere. Concerning the possible generalizations to higher dimensions, A. S. Besicovitch proved the following: there exists a continuous function of two variables such that at each point of a set of positive measure there exist continuum many directions, in each of which one Dini derivative is infinite and the other three are zero, thus violating the bilateral Denjoy property.
Our aim is to show that for two-variable continuous functions it is possible that on a set of positive measure there exist directions in which even the one-sided Denjoy behaviour is violated. We construct continuous functions of two variables such that (i) both of its one-sided derivatives equal ∞ in continuum many directions on a set of positive measure, and (ii) all four directional Dini derivatives are finite and distinct in continuum many directions on a set of positive measure.

Kategorie tematyczne

• 26B05: Continuity and differentiation questions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 141
• Numer: 2
• Strony: 157-173

Daty

wydano

2015

Twórcy

autor

Miklós Laczkovich

• Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C, 1117 Hungary

autor

Ákos K. Matszangosz

• Department of Mathematics and its Applications, Central European University, Budapest, Nádor utca 9, 1051 Hungary

Identyfikatory

DOI

10.4064/cm141-2-2