We comment on a problem of Mazur from “The Scottish Book" concerning second partial derivatives. We prove that if a function f(x,y) of real variables defined on a rectangle has continuous derivative with respect to y and for almost all y the function has finite variation, then almost everywhere on the rectangle the partial derivative exists. We construct a separately twice differentiable function whose partial derivative is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-2-3,
author = {Volodymyr Mykhaylyuk and Anatolij Plichko},
title = {On a problem of Mazur from "The Scottish Book" concerning second partial derivatives},
journal = {Colloquium Mathematicae},
volume = {139},
year = {2015},
pages = {175-181},
zbl = {1339.26029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-2-3}
}
Volodymyr Mykhaylyuk; Anatolij Plichko. On a problem of Mazur from "The Scottish Book" concerning second partial derivatives. Colloquium Mathematicae, Tome 139 (2015) pp. 175-181. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm141-2-3/