Definition:Continuous Complex Function/Using Limit

Definition
Let $A_1, A_2 \subseteq \C$ be subsets of the complex plane.

Let $f: A_1 \to A_2$ be a complex function from $A_1$ to $A_2$.

Let $a \in A_1$.

$f$ is continuous at (the point) $a$ :
 * The limit of $\map f z$ as $z \to a$ exists, and
 * $\ds \lim_{z \mathop \to a} \map f z = \map f a$