On a problem of Mazur from "The Scottish Book" concerning second partial derivatives

Volodymyr Mykhaylyuk; Anatolij Plichko

Colloquium Mathematicae (2015)

  • Volume: 141, Issue: 2, page 175-181
  • ISSN: 0010-1354

Abstract

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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 F y ( x ) : = f y ' ( x , y ) has finite variation, then almost everywhere on the rectangle the partial derivative f y x ' ' exists. We construct a separately twice differentiable function whose partial derivative f x ' is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.

How to cite

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Volodymyr Mykhaylyuk, and Anatolij Plichko. "On a problem of Mazur from "The Scottish Book" concerning second partial derivatives." Colloquium Mathematicae 141.2 (2015): 175-181. <http://eudml.org/doc/283514>.

@article{VolodymyrMykhaylyuk2015,
abstract = {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 $F_\{y\}(x): = f^\{\prime \}_\{y\}(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f^\{\prime \prime \}_\{yx\}$ exists. We construct a separately twice differentiable function whose partial derivative $f^\{\prime \}_\{x\}$ is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.},
author = {Volodymyr Mykhaylyuk, Anatolij Plichko},
journal = {Colloquium Mathematicae},
keywords = {mixed derivative; differentiability; measurability},
language = {eng},
number = {2},
pages = {175-181},
title = {On a problem of Mazur from "The Scottish Book" concerning second partial derivatives},
url = {http://eudml.org/doc/283514},
volume = {141},
year = {2015},
}

TY - JOUR
AU - Volodymyr Mykhaylyuk
AU - Anatolij Plichko
TI - On a problem of Mazur from "The Scottish Book" concerning second partial derivatives
JO - Colloquium Mathematicae
PY - 2015
VL - 141
IS - 2
SP - 175
EP - 181
AB - 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 $F_{y}(x): = f^{\prime }_{y}(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f^{\prime \prime }_{yx}$ exists. We construct a separately twice differentiable function whose partial derivative $f^{\prime }_{x}$ is discontinuous with respect to the second variable on a set of positive measure. This solves the Mazur problem in the negative.
LA - eng
KW - mixed derivative; differentiability; measurability
UR - http://eudml.org/doc/283514
ER -

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