Identity Theorem

Theorem
Let $U$ be an open,  connected subset of $\C$.

Let $f$ and $g$ be mappings on $U$.

Let $S = \left\{{z \in U : f(z) = g(z)}\right\}$.

Suppose that $f$ and $g$ are analytic on $U$ and that $S$ has a  limit point in $U$.

Then:
 * $\forall z \in U : f(z) = g(z)$