Definition:Boundary Point (Complex Analysis)

Definition
Let $S \subseteq \C$ be a subset of the complex plane.

Let $z_0 \in \C$.

$z_0$ is a boundary point of $S$ iff every $\epsilon$-neighborhood $N_{z_0}$ of $z_0$ contains points of $\C$ in $S$ and also points of $\C$ which are not in $S$.