Let $\Omega\subseteq\Bbb C^2$ be open bounded (and connected), $f:\Omega\to\Bbb C$ separately holomorphic (i.e. $f$ is holomorphic in each variable when the other is fixed).
Hartogs theorem is not allowed, so we can't say $f$ is holomorphic on $\Omega$.
Now I think that in this case $f$ is continous in $\bar\Omega$ iff $f$ is integrable in $\Omega$. In other words $f\in\mathcal{C}^0(\bar\Omega)\Leftrightarrow f\in L^1(\Omega)$.
Clearly one implication ($\Rightarrow$) is always true. But the other?
At the moment I can't get no proof nor counterexamples. Can someone help me? Many thanks