Identity Theorem

Theorem
Let $U$ be an open connected subset of the complex plane $\C$.

Let $f$ and $g$ be complex functions whose domain is $U$.

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

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

Let $S$ have a limit point in $U$.

Then:
 * $\forall z \in U : f \left({z}\right) = g \left({z}\right)$