Let $f : \mathbb R \times \mathbb R \to \mathbb R$ be continuous when we fix one variable. Then $f$ need not be continuous (see e.g. Functions continuous in each variable ).
Does it imply that $f$ is locally bounded?
I would be surprised, but couldn't immediately think of a counterexample.
Context: I'm interested in $f : \mathbb R \times \mathbb C \to \mathbb C$ that are continuous in the first and analytic in the second variable. In this case, Cauchy's integral formula + Dominated convergence tells us that locally bounded implies jointly continuous.