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2015 | 141 | 2 | 157-173
Tytuł artykułu

On the linear Denjoy property of two-variable continuous functions

Treść / Zawartość
Warianty tytułu
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.
Słowa kluczowe
Rocznik
Tom
141
Numer
2
Strony
157-173
Opis fizyczny
Daty
wydano
2015
Twórcy
  • Department of Analysis, Eötvös Loránd University, Budapest, Pázmány Péter sétány 1/C, 1117 Hungary
  • Department of Mathematics and its Applications, Central European University, Budapest, Nádor utca 9, 1051 Hungary
Bibliografia
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-2-2
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