Definition:Continuous Complex Function

Definition
As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.

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 $$a \in A_1$$ be a point in $$A_1$$.

Definition using Limit
$$f$$ is continuous at (the point) $$a$$ iff:
 * The limit of $$f \left({z}\right)$$ as $$z \to a$$ exists;
 * $$\lim_{z \to a} f \left({z}\right) = f \left({a}\right)$$.

Epsilon-Delta Definition
$$f$$ is continuous at (the point) $$a$$ iff:


 * $$\forall \epsilon > 0: \exists \delta > 0: \left|{z - a}\right| < \delta \implies \left|{f \left({z}\right) - f \left({a}\right)}\right| < \epsilon$$.

Epsilon-Neighborhood Definition
$$f$$ is continuous at (the point) $$a$$ iff:
 * $$\forall N_\epsilon \left({f \left({a}\right)}\right): \exists N_\delta \left({a}\right): f \left({ N_\delta \left({a}\right)}\right) \subseteq N_\epsilon \left({f \left({a}\right)}\right)$$.

where $$N_\epsilon \left({a}\right)$$ is the $\epsilon$-neighborhood of $$a$$ in $$M_1$$.

That is, for every $$\epsilon$$-neighborhood of $$f \left({a}\right)$$ in $$\C$$, there exists a $$\delta$$-neighborhood of $$a$$ in $$\C$$ whose image is a subset of that $$\epsilon$$-neighborhood.

Open Set Definition
$$f$$ is continuous iff:
 * for every set $$U \subseteq \C$$ which is open in $$\C$$, $$f^{-1} \left({U}\right)$$ is open in $$\C$$.