Definition:Limit of a 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

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

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

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

where:

• $N_\delta \left({c}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$;
• $N_\epsilon \left({L}\right)$ is the $\epsilon$-neighborhood of $L$ in $M_2$.

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.

Notes

1. $c$ does not need to be a point in $A_1$. Therefore $f \left({c}\right)$ need not be defined. And even if $c \in A_1$, in may be that $f \left({c}\right) \ne L$.
2. It is essential that $c$ be a limit point of $A_1$. Otherwise there would exist $\delta > 0$ such that $\left\{{z: 0 < \left|{z - c}\right| < \delta}\right\}$ contains no points of $A_1$. In this case the first condition would be vacuously true for any $L \in A_2$, which would not do.