Definition:Limit of Complex Function

Definition
The definition for the limit of a complex function is exactly the same as that for the general metric space.

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

Let $$c$$ be a limit point of $$A_1$$.

Let $$f: A_1 \to A_2$$ be a complex function from $$A_1$$ to $$A_2$$ defined everywhere on $$A_1$$ except possibly at $$c$$.

Let $$L \in A_2$$.

Then $$f \left({z}\right)$$ is said to tend to the limit $$L$$ as $$z$$ tends to $$c$$, and we write:
 * $$f \left({z}\right) \to L$$ as $$z \to c$$

or
 * $$\lim_{z \to c} f \left({z}\right) = L$$

if the following equivalent conditions hold.

This is voiced "the limit of $$f \left({z}\right)$$ as $$z$$ tends to $$c$$".

Epsilon-Delta Condition

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

where $$\delta, \epsilon \in \R$$.

That is, for every real positive $$\epsilon$$ there exists a real positive $$\delta$$ such that every point in the domain of $$f$$ within $$\delta$$ of $$c$$ has an image within $$\epsilon$$ of some point $$L$$ in the codomain of $$f$$.

Epsilon-Neighborhood Condition

 * $$\forall N_\epsilon \left({L}\right): \exists N_\delta \left({c}\right) - \left\{{c}\right\}: f \left({N_\delta \left({c}\right) - \left\{{c}\right\}}\right) \subseteq N_\epsilon \left({L}\right)$$.

where:
 * $$N_\delta \left({c}\right) - \left\{{c}\right\}$$ is the deleted $\delta $-neighborhood of $$c$$ in $$M_1$$;
 * $$N_\epsilon \left({f \left({c}\right)}\right)$$ is the $\epsilon$-neighborhood of $$a$$ in $$M_1$$.

That is, for every $$\epsilon$$-neighborhood of $$L$$ in $$A_2$$, there exists a deleted $$\delta$$-neighborhood of $$c$$ in $$A_1$$ whose image is a subset of that $$\epsilon$$-neighborhood.

Equivalence of Definitions
These definitions are seen to be equivalent by the definition of the $\epsilon$-neighborhood.