Let $f:[0,1]\times\mathbb R\to\mathbb R$. For $r>0$ consider the estimate \begin{align*} |f(s,u)-f(s,v)-f(t,u)+f(t,v)|\leq L_r|s-t||u-v|,\tag{$\star$} \end{align*} for all $s,t\in[0,1]$ and $u,v\in[-r,r]$, where $L_r>0$ is a constant only depending on $r$.
My question is: How does $f$ have to look like to satisfy $(\star)$ for each $r>0$?
Clearly, if $f$ is in separated variables, that is, $f$ has the form $f(t,u)=g(t)h(u)$, then $f$ satisfies $(\star)$, if and only if $g\in Lip[0,1]$ and $h\in Lip[-r,r]$ for each $r>0$. But are there other functions $f$ satisfying $(\star)$ which are not in separated variables? For example, does each $f\in C^\infty$ satisfy $(\star)$? I tried $f(t,u)=e^{tu}$ but was not able to decide whether $(\star)$ is fulfilled.
Any help is highly appreciated. Thanks in advance!