Picard's great theorem

Statement

Let $c \in \mathbb{C}$, let $r > 0$, and let $f$ be holomorphic on the punctured disc $0 < |z - c| < r$. Assume that $c$ is an essential singularity of $f$. Then for each $\delta$ with $0 < \delta < r$, there is at most one $w_0 \in \mathbb{C}$ with the following property. The function $f$ takes the value $w_0$ only finitely many times on $0 < |z - c| < \delta$. Thus, on each punctured disc around $c$, $f$ takes each complex value infinitely often, with at most one exception.

Remarks

The exception can occur: $e^{1/z}$ never takes the value $0$, and it takes each $w \neq 0$ infinitely often near $0$, see exp(1/z). The theorem is much stronger than the Casorati–Weierstrass theorem, which gives only that the values come close to each $w$. Picard's little theorem is a related result: a nonconstant entire function takes all complex values with at most one exception.

Explanations

L1 human

Near an essential singularity, the function does not only come close to each value: it hits almost every value infinitely many times. At most one value can be missed, as $0$ is missed by $e^{1/z}$.