Suppose $f$ is bounded and analytic on the open unit disk $\mathbb{D}$. Say that $f$ extends continuously to one point $z_0$ on $\partial \mathbb{D}$, the boundary of $\mathbb{D}$. Now does the maximum principle apply here, i.e., is it true that $|f(z_0)| \le M$ implies $|f(z)| \le M$ for $z \in \mathbb{D}$? (Equivalently, $|f(z)| \le |f(z_0)|$.)
I know it sounds strange to talk about continuous extensions to just one point on the boundary. This question is derived from Gamelin's Complex Analysis, p. 89, #7. In that problem, there are finitely many points on the boundary such that $f$ extends continuously to the arcs between them. I'm wondering if this is true when there's only one such point.
(I understand that analytic continuation along the whole boundary might be possible, but I haven't learned about that yet.)