On the linear Denjoy property of two-variable continuous functions

Miklós Laczkovich, and Ákos K. Matszangosz. "On the linear Denjoy property of two-variable continuous functions." Colloquium Mathematicae 141.2 (2015): 157-173. <http://eudml.org/doc/286663>.

@article{MiklósLaczkovich2015,
abstract = { 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. },
author = {Miklós Laczkovich, Ákos K. Matszangosz},
journal = {Colloquium Mathematicae},
keywords = {Denjoy-Young-Saks theorem; two-variable functions},
language = {eng},
number = {2},
pages = {157-173},
title = {On the linear Denjoy property of two-variable continuous functions},
url = {http://eudml.org/doc/286663},
volume = {141},
year = {2015},
}

TY - JOUR
AU - Miklós Laczkovich
AU - Ákos K. Matszangosz
TI - On the linear Denjoy property of two-variable continuous functions
JO - Colloquium Mathematicae
PY - 2015
VL - 141
IS - 2
SP - 157
EP - 173
AB - 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.
LA - eng
KW - Denjoy-Young-Saks theorem; two-variable functions
UR - http://eudml.org/doc/286663
ER -