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

• Autorzy: Volodymyr Mykhaylyuk , Anatolij Plichko
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 26B30: Absolutely continuous functions, functions of bounded variation
• 26B05: Continuity and differentiation questions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2015
• Tom: 141
• Numer: 2
• Strony: 175-181

Daty

wydano

2015

Twórcy

autor

Volodymyr Mykhaylyuk

• Department of Applied Mathematics, Chernivtsi National University, 58-012 Chernivtsi, Ukraine

autor

Anatolij Plichko

• Institute of Mathematics, Cracow University of Technology, 31-155 Kraków, Poland

Identyfikatory

DOI

10.4064/cm141-2-3