# Zeros of analytic function and limit points at boundary

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).

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what are the zeros of your function?where they are in complex plane? – Un Chien Andalou Feb 13 '13 at 4:28
You need to add some more details, but the idea is sound. Nitpicking: you mean domain or open set, not open subspace. Also, the exact answer depends on what you mean by cluster: do you want the zeros to have one accumulation point at the boundary, or do you want the accumulation points to be the entire boundary? (If it's the second case, you need to take another function.) – mrf Feb 13 '13 at 10:22