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

Volume 141 / 2015

Volodymyr Mykhaylyuk, Anatolij Plichko Colloquium Mathematicum 141 (2015), 175-181 MSC: 26B05, 26B30. DOI: 10.4064/cm141-2-3

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 $F_y(x):=f'_y(x,y)$ has finite variation, then almost everywhere on the rectangle the partial derivative $f''_{yx}$ 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.

Authors

  • Volodymyr MykhaylyukDepartment of Applied Mathematics
    Chernivtsi National University
    58-012 Chernivtsi, Ukraine
    e-mail
  • Anatolij PlichkoInstitute of Mathematics
    Cracow University of Technology
    31-155 Kraków, Poland
    e-mail

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