Identity Theorem

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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:

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


Proof