Definition:Limit of Function (Metric Space)

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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 $L \in M_2$.


$f \left({x}\right)$ is said to tend to the limit $L$ as $x$ tends to $c$ and is written:

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

or

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

if and only if the following equivalent conditions hold:


$\epsilon$-$\delta$ Condition

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

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 $f$.


$\epsilon$-Ball Condition

$\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left({B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}}\right) \subseteq B_\epsilon \left({L; d_2}\right)$.

where:

$B_\delta \left({c; d_1}\right) \setminus \left\{{c}\right\}$ is the deleted $\delta $-neighborhood of $c$ in $M_1$
$B_\epsilon \left({L; d_2}\right)$ is the open $\epsilon$-ball of $L$ in $M_2$.


That is, for every open $\epsilon$-ball of $L$ in $M_2$, there exists a deleted $\delta $-neighborhood of $c$ in $M_1$ whose image is a subset of that open $\epsilon$-ball.


This is voiced:

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


Equivalence of Definitions

These definitions are seen to be equivalent in Equivalence of Definitions of Limit of Function in Metric Space.


Also known as

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

can also be voiced as:

$f \left({x}\right)$ approaches the limit $L$ as $x$ approaches $c$