Metric Space Continuity by Epsilon-Delta

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

Let $f: A_1 \to A_2$ be a mapping from $A_1$ to $A_2$.

Let $a \in A_1$ be a point in $A_1$.

Then the following definitions of continuity of $f$ at $a$ with respect to $d_1$ and $d_2$ are equivalent:

Definition by Limits implies $\epsilon$-$\delta$ Definition
Suppose that $f$ is $\left({d_1, d_2}\right)$-continuous at $a$ in the sense that:
 * $(1): \quad$ The limit of $f \left({x}\right)$ as $x \to a$ exists
 * $(2): \quad \displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.

Let $\displaystyle L = \lim_{x \to a} f \left({x}\right)$.

Then by the $\epsilon$-$\delta$ definition of limit:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: 0 < d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), L}\right) < \epsilon$

By $(2)$:
 * $\displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$

that is:
 * $f \left({a}\right) = L$

Then:
 * $0 = d_1 \left({a, a}\right)$

and so:
 * $d_2 \left({f \left({a}\right), L}\right) = 0 < \epsilon$

Thus the definition by limits implies the $\epsilon$-$\delta$ definition.

$\epsilon$-$\delta$ Definition implies Definition by Limits
Suppose that $f$ is $\left({d_1, d_2}\right)$-continuous at $a$ in the sense that:
 * $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: d_1 \left({x, a}\right) < \delta \implies d_2 \left({f \left({x}\right), f \left({a}\right)}\right) < \epsilon$

where $\R_{>0}$ denotes the set of all strictly positive real numbers.

There are two possibilities for $d_1 \left({x, a}\right)$:

$(A): \quad 0 < d_1 \left({x, a}\right) < \delta$

Then by the $\epsilon$-$\delta$ definition of limit:
 * $\displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$

$(B): \quad d_1 \left({x, a}\right) = 0$

That is, $x = a$.

Hence:
 * $f \left({x}\right) = f \left({a}\right)$

and so again:
 * $\displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.

Thus the $\epsilon$-$\delta$ definition implies the definition by limits.

Also see

 * Metric Space Continuity by Open Ball