Continuity of function which is Lipschitz with respect to each variables separately Let a function $f: I\times J \rightarrow \mathbb R$, where $I,J$ are intervals in $\mathbb R$, be Lipschitz with respect of each variable separately. Is it then $f$ continuous with respect of both variables?
Thanks

It depends on what you mean with Lipschitz with respect to each variable separately.
Consider the function $f\colon\mathbb R^2\to \mathbb R$, $(x,y)\mapsto xy$.
Then for fixed $y$, the function $x\mapsto f(x,y)$ is Lipschitz (with Lipschitz constant $y$ depending on $y$) and similarly $y\mapsto f(x,y)$ for fixed $x$.
However $f$ itself is not Lipschitz as $f(t,t)=t^2$ shows.
On the other hand, if there is a single Lipschitz constant $L$ working for all functions of the form $x\mapsto f(x,y)$ and $y\mapsto f(y,x)$, then $f$ is Lipschitz with constant $L\sqrt 2$.
