Let $f \in \mathcal{O}(D) \cap \mathcal{C}(\overline{D})$, $D$ - a bounded region in $\mathbb{C}$. Suppose that $f(z)=0$ for $z \in \partial D$.
Prove that $f(z)=0$ for all $z \in D$.
At first I thought about identity principle, but there we require that $f=g$ on a set $A$ with accumulation point in $int D$ .
And Poisson formula works for a circle - here we have an arbitrary closed region.
Could you tell me what I could use to prove the statement above?
The real and imaginary parts are continuous so they reach a maximum. If the maximum is nonzero it must be reached in the interior.