On the mixed derivatives of a separately twice differentiable function

TL;DR

This paper proves that functions with square integrable second derivatives in each variable almost everywhere possess mixed derivatives, advancing understanding of differentiability properties in multivariable calculus.

Contribution

It establishes the almost everywhere existence of mixed derivatives for functions with square integrable second derivatives, a novel result in the theory of multivariable functions.

Findings

Mixed derivatives exist almost everywhere under given conditions

Square integrability of second derivatives implies mixed derivatives almost everywhere

Advances understanding of differentiability in multivariable calculus

Abstract

We prove that a function $f(x,y)$ of real variables defined on a rectangle, having square integrable partial derivatives $f"_{xx}$ and $f"_{yy}$, has almost everywhere mixed derivatives $f"_{xy}$ and $f"_{yx}$.