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

Volodymyr Mykhaylyuk ¹   ; Anatolij Plichko ²

¹ Department of Applied Mathematics Chernivtsi National University 58-012 Chernivtsi, Ukraine
² Institute of Mathematics Cracow University of Technology 31-155 Kraków, Poland

Colloquium Mathematicum, Tome 141 (2015) no. 2, pp. 175-181

Résumé

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.

Détail

DOI : 10.4064/cm141-2-3

Keywords: comment problem mazur scottish book concerning second partial derivatives prove function real variables defined rectangle has continuous derivative respect almost function y has finite variation almost everywhere rectangle partial derivative exists construct separately twice differentiable function whose partial derivative discontinuous respect second variable set positive measure solves mazur problem negative