A complex valued continuous function which is holomorphic outside of its zeros Let $D$ be a non-empty connected open subset of $\mathbb{C}^n$.
Let $f$ be a complex valued continuous function on $D$.
Let $Z$ be the set of zeros of $f$.
Suppose $f$ is holomoprphic on $D - Z$.
Is $f$ holomorphic on $D$?

I think the answer to your question is yes. There's probably a better answer than this, but I think the following argument should work.
For each $M\in \mathbb{R}$, let $\varphi_M\colon D\to \mathbb{R}$ be the function $\varphi_M(z) = \max(M, \log|f(z)|)$. This function is plurisubharmonic on $D$, and $\lim_{M\to -\infty} \varphi_M(z) = \log |f(z)|$ for each $z\in D$. Since decreasing limits of plurisubharmonic functions are plurisubharmonic, we conclude the function $\log|f(z)|$ is plurisubharmonic on $D$. In particular, $Z$ is a pluripolar set. One can then use the following result to conclude that $f$ is holomorphic in $D$.

Let $D\subset\mathbb{C}^n$ be open, and let $Z$ be a closed pluripolar subset of $D$. Suppose $f$ is holomorphic on $D\smallsetminus Z$. If $f$ is locally bounded at every point in $Z$, then $f$ extends to a holomorphic function on $D$.

For a reference, see Corollary 5 of Chapter Q of Volume 1 of Introduction to holomorphic functions of several variables by R.C. Gunning.
