On a problem of Mazur from ‶The Scottish Book″ concerning second partial derivatives
Volume 141 / 2015
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.
