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

摘要

We comment on a Mazur problem from "Scottish Book" concerning second partialderivatives. It is proved that, if a function $f(x,y)$ of real variablesdefined on a rectangle has continuous derivative with respect to $y$ and foralmost all $y$ the function $\,F_y(x):=f'_y(x,y)$ has finite variation, thenalmost everywhere on the rectangle there exists the partial derivative$f"_{yx}$. We construct a separately twice differentiable function, whosepartial derivative $f'_x$ is discontinuous with respect to the second variableon a set of positive measure. This solves in the negative the Mazur problem.