Definition:Limit of a Function (Metric Space)

Definition

Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ be metric spaces.

Let $c$ be a limit point of $M_1$.

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

Let $L \in M_2$.

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

$f \left({x}\right) \to L$ as $x \to c$

or

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

if the following equivalent conditions hold.

This is voiced:

the limit of $f \left({x}\right)$ as $x$ tends to $c$.

Epsilon-Delta Condition

$\forall \epsilon > 0: \exists \delta > 0: 0 < d_1 \left({x, c}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$

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 $M_2$, there exists a deleted $\delta$-neighborhood of $c$ in $M_1$ 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 $\left\{{z: 0 < d_1 \left({z, c}\right) < \delta}\right\}$ 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.