Here is the problem statement:
Suppose $f$ is a holomorphic function on the unit disk. Show that the set $A=\lbrace (z,w) \in \mathbb{C}^2\;|\; |z|,|w| \leq \frac{1}{2}, z\neq w, f(z)=f(w)\rbrace$ is either finite or uncountably infinite.
I'm pretty stuck, but here are some thoughts:
The bound on the $z,w$ is a ltitle odd. The only thing I take away from it is that $|z+w| \leq |z| + |w| = 1$ so the sum stays in the closure of the disk. But this doesn't seem relevant.
I tried to find a clever way to use the identity principle but I couldn't. For example, suppose to the contrary the set $A$ is countably infinite, then it is pointless to consider the function $g(z) = f(z) - f(w_0)$ for some $w_0 \in A$ since there doesn't need to be countably many $z_0$ matching up with $w_0$ in the sense $f(z_n) = f(w_0)$, only countably many pairs $z,w$ with the same images.
I tried representing $f(z)$ as a power series centered at $z_0=0$: $ f(z) = \sum_{n=0}^\infty a_nz^n $ and then considering expressions like $ f(z_0) - f(w_0) = \sum_{n=0}^\infty a_n(z_0^n - w_0^n) $ to learn something about the coefficients but this didn't lead anywhere.
I think there must be some slick solution but I can't see it, so I'd prefer hints rather than full solutions right now.