This nice paper by Gilles Godefroy (in French) tells us the story of Baire's lemma. In 1896, Monsieur Baire was lecturing on analysis and carelessly gave the following exercise: to find all solutions to the differential equation $$\tag{1} {\partial f \over \partial x} + {\partial f \over \partial y} =0. $$ This is rather easy if $f$ is assumed to be differentiable, for in that case the equation can be rewritten as $$ {\partial f \over \partial \vec{v}}=0,\qquad \vec{v}=(1, 1),$$ where $\partial/\partial \vec{v}$ denotes directional derivative. From this one concludes that $$\tag{2}f(x, y)=\phi(x-y)$$ for an arbitrary differentiable function $\phi$ of one variable.
But what if $f$ is not differentiable?
According to the paper, this train of thoughts led Baire to investigate the subject of separately continuous functions. Those are functions of two or more variables that are continuous if all variables minus one are held fixed. (Indeed, for equation (1) to hold, partial derivatives need at least exist, and so $f$ must be separately continuous). This is very interesting, but unfortunately the paper does not tell the end of the story about the differential equation! Hence the question.
Question. Do there exist solutions to the equation (1) that are not of the form (2)?
Here a solution is a function $f\colon \mathbb{R}^2\to \mathbb{R}$ that admits partial derivatives with respect to both variables and that satisfies relation (1) at all points.