Holomorphic function having finitely many zeros in the open unit disc Suppose $f$ is continuous on the closed unit disc $\overline{\mathbb{D}}$ and is holomorphic on the open unit disc $\mathbb{D}$. Must $f$ have finitely many zeros in $\mathbb{D}$? I know that this is true if $f$ is holomorphic in $\overline{\mathbb{D}}$ (by compactness of the closed unit disc), but I'm not sure of what happens when I just consider $\mathbb{D}$.

Any accumulation point of the zeros must be on the boundary, of course.  You can show in this case that the series centered at 0 has radius of convergence 1.  For a specific example, I think the following works:
For $|z| \leq 1$, $$f(z) = (z-1)\prod_{n=1}^\infty\frac{z-1+1/n^2}{1-(1-1/n^2)z}.$$
