Definition:Limit of Mapping between Metric Spaces

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 the limit $$L$$ as $$x$$ tends to $$c$$, and we write:
 * $$f \left({x}\right) \to L$$ as $$x \to c$$

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

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

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

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 $$M_2$$, 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.