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

Abstract

We comment on a Mazur problem from "Scottish Book" concerning second partial derivatives. It is proved 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 $F_y(x):=f_y^{\prime }(x,y)$ has finite variation, then almost everywhere on the rectangle there exists the partial derivative $f{"}_{yx}$. We construct a separately twice differentiable function, whose partial derivative $f_x^{\prime }$ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur problem.