Suppose an analytic function $f$ has an essential singularity at $z=a$. Then in each neighbourhood of $a$, $f$ assumes each complex number, with one possible exception, an infinite number of times.
This is the statement of Great Picard Theorem from Function of One Complex Variable by Conway. I am unable to find the version of Great Picard Theorem which is written mathematically in symbol.
What I am confused is the phrase in bold. The following two options are my guess but I am not sure which one is correct:
(i) for every $w$ in the neighbourhood of $a$, there are infinitely many $z\in \mathbb{C}$ such that $f(z)=w$
(ii) for every $w\in \mathbb{C}$, there are infinitely many $z$ in the neighbourhood of $a$ such that $f(z)=w$
It's the second possibility. And don't forget the “with one possible exception” part. Think about $e^{1/z}$, for instance.