On the linear Denjoy property of two-variable continuous functions

Miklós Laczkovich ¹   ; Ákos K. Matszangosz ²

¹ Department of Analysis Eötvös Loránd University Budapest, Pázmány Péter sétány 1/C 1117 Hungary
² Department of Mathematics and its Applications Central European University Budapest, Nádor utca 9 1051 Hungary

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 157-173

Résumé

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 $\infty $ 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.

Détail

DOI : 10.4064/cm141-2-2

Keywords: classical denjoy young saks theorem gives relation here termed denjoy property between dini derivatives arbitrary one variable function holds almost everywhere concerning possible generalizations higher dimensions nbsp nbsp besicovitch proved following there exists continuous function variables each point set positive measure there exist continuum many directions each which dini derivative infinite other three zero violating bilateral denjoy property two variable continuous functions possible set positive measure there exist directions which even one sided denjoy behaviour violated construct continuous functions variables its one sided derivatives equal infty continuum many directions set positive measure directional dini derivatives finite distinct continuum many directions set positive measure