Almost everywhere matters

Abstract

Suppose that $F$ is a homeomorphism of ${ℝ}^2$ that is differentiable almost everywhere. If $F\left(x,y\right)=\left(f\left(x\right),g\left(y\right)\right)$, then it is clear that the derivative of $F$ is given by a diagonal matrix when it exists. Is the converse true? We explain how the distinction between differentiable everywhere and differentiable almost everywhere is important in this question.