Question: Let $D\subset \mathbb{C}^N$ for $N\geq 2$ be open, bounded and connected and let $f: \overline{D} \rightarrow \mathbb{C}$ be continuous such that $f$ is holomorphic on $D$. Show that if $f$ has a zero on $D$, then $f$ has a zero on $\partial D$.
I'm not 100% sure how start this. I was thinking of maybe trying to use the maximum modulus principle to arrive at a contradiction but I don't really get anywhere.
Any help is much appreciated.
Answered my own question, had use that the the zero set for $N\geq 2$ has no isolated points, but that if $Z(f)$ doesn't intersect the boundary, it is clearly bounded. Hence it is compact and therefore finite, leading to a contradiction :)