Let $f: U \rightarrow \mathbb{C}$ where $U$ is open in $\mathbb{C}^n$ and suppose every coordinate cross section of $f$ is holomorphic (I hope that's not too colloquial).
I've heard it's a somewhat deep theorem that this is enough to imply f is continuous, and therefore holomorphic as a function of n variables. However when one uses Cauchy's formula (in one variable) to get a power series expansion of $f$ in a neighborhood of a point, one seems only to use that $f$ is locally bounded (so one can bound the values of f on the distinguished boundary of a polydisc). Is that correct? I.e. does {locally bounded} + {holomorphic cross sections} so easily imply holomorphic or have I made a mistake?
Yes, if $f$ is locally bounded and separately holomorphic, it's not hard to see that $f$ is in fact continuous and jointly analytic. See this question for more details.
As zwh pointed out, $f$ being separately holomorphic is enough, but this is a lot more difficult to prove.