I am wondering if the following classes of functions have a name and if they play a role in some branch of analysis. For $L\geq0$, define $C_L$ as $$C_L = \{f:\mathbb R^2\to \mathbb R: |f(x,y)-f(x',y)+f(x',y')-f(x,y')|\leq L|x-x'||y-y'|\}\,.$$
Note that if $f$ depends only on $x$ (or only on $y$), then it is in $C_0$, and so it il in $C_L$ for all $L$. This means that $f$ can be extremely irregular and yet be in $C_0$.
If $\partial x\partial y f$ exists everywhere then the fact that it is everywhere bounded by $L$ is equivalent to $f$ being in $C_L$.
An elementary property of these functions is that if $f\in C_L$ and for some $y$ we have that $x\mapsto f(x, y)$ is $M$-Lipschitz, then we have that $x\mapsto f(x, y')$ is $M+L|y-y'|$-Lipschitz for all $y'$.
Note that $f$ being Lipschitz is not enough for this property to be true. For example $f(x, y)=|x-y|$ is not in $C_L$ for any $L$.