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

TL;DR

This paper investigates a problem from 'The Scottish Book' about second partial derivatives, proving conditions for their existence and constructing a counterexample that shows the problem's negative resolution.

Contribution

It establishes conditions under which the mixed second partial derivative exists and provides a counterexample to Mazur's problem, demonstrating the derivative can be discontinuous.

Findings

Proved existence of mixed second derivatives under certain conditions.

Constructed a counterexample with discontinuous partial derivative.

Solved Mazur's problem negatively.

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 $\,F_y(x):=f'_y(x,y)$ 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 $f'_x$ is discontinuous with respect to the second variable on a set of positive measure. This solves in the negative the Mazur problem.