Are two analytic functions equal if they are equal on the boundary of an open disk?

Question

Let $D$ be the open disk in $\mathbb{C}$ with origin $0$ and radius $1$.

Let $f,g: \overline{D} \to \mathbb{C}$ be continuous functions such that $f$ and $g$ are analytic on $D$ and such that $f=g$ on $S^1= \{z \in \mathbb{C}: |z| = 1\}$. Can I conclude that $f=g$ on $D$ as well?

It is enough to show that $\{z\in D: f(z) = g(z)\}$ has a limit point in $D$ but I can't see why this should hold.

Answer 1

Suppose that $f\neq g$. Let $M=\max_{z\in\overline D}\bigl\lvert f(z)-g(z)\bigr\rvert$, which is greater than $0$. Then there is some $z_0\in\overline D$ such that $\bigl\lvert f(z_0)-g(z_0)\bigr\rvert=M$. Then $z_0\in D$ since, if $z_0\in S^1$, then $f(z_0)-g(z_0)=0$. Now, apply the maximum modulus principle to deduce that $f=g$.