Let $S$ be the open ball of center $0$ and radius $1$ with $0$ removed in the complex plane. Is the function $f(z)=\sin(1/z)$ a valid example of analytic function defined in an open subspace whose limit points of zeros cluster at the boundary?
EDIT
For open subspace I simply mean open set.
Existence of limit points for zeros of $f$ is meant in a larger sense, not restrictive as asking for infinte limit points on the boundary, (however, I am really interested in the latter case too).