I'm reading through a proof on the uniqueness theorem of Laurent series in my lecture notes and came across this sentence:
The point $c$ ($\in \mathbb{C})$ is a limit point of zero points of the (analytic) function $f$ and therefore there exists a circle $K$ around $c$ where $f=0$.
Under what circumstances is this true? It seems to me it should not be true generally (what if the zero points only were on one side?) but I don't see any context that would give sufficient assumptions (nor know what they would be).
Thank you.
The general statement is that if $f$ is analytic on a (connected) domain $D$ and not constant there, then the zeros of $f$ have no limit point in $D$. So if $f$ is supposed to be analytic in some neighbourhood of $c$, $f$ is identically $0$ in that neighbourhood.