On the linear Denjoy property of two-variable continuous functions

Volume 141 / 2015

Miklós Laczkovich, Ákos K. Matszangosz Colloquium Mathematicum 141 (2015), 157-173 MSC: Primary 26B05. DOI: 10.4064/cm141-2-2


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.


  • Miklós LaczkovichDepartment of Analysis
    Eötvös Loránd University
    Budapest, Pázmány Péter sétány 1/C
    1117 Hungary
  • Ákos K. MatszangoszDepartment of Mathematics and its Applications
    Central European University
    Budapest, Nádor utca 9
    1051 Hungary

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