# Equivalence of Definitions of Metric Space Continuity at Point

## 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 $A_1$.

The following definitions of the concept of **Continuous at Point of Metric Space** are equivalent:

### $\epsilon$-$\delta$ Definition

**$f$ is continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in A_1: \map {d_1} {x, a} < \delta \implies \map {d_2} {\map f x, \map f a} < \epsilon$

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

### Definition by Limits

$f$ is **continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)** if and only if:

- $(1): \quad$ The limit of $f \left({x}\right)$ as $x \to a$ exists
- $(2): \quad \displaystyle \lim_{x \to a} f \left({x}\right) = f \left({a}\right)$.

### $\epsilon$-Ball Definition

$f$ is **continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: f \left[{B_\delta \left({a; d_1}\right)}\right] \subseteq B_\epsilon \left({f \left({a}\right); d_2}\right)$

where $B_\epsilon \left({f \left({a}\right); d_2}\right)$ denotes the open $\epsilon$-ball of $f \left({a}\right)$ with respect to the metric $d_2$, and similarly for $B_\delta \left({a; d_1}\right)$.

### Definition by Neighborhood

$f$ is **continuous at (the point) $a$ (with respect to the metrics $d_1$ and $d_2$)** if and only if:

- for each neighborhood $N'$ of $f \left({a}\right)$ in $M_2$ there exists a corresponding neighborhood $N$ of $a$ in $M_1$ such that $f \left[{N}\right] \subseteq N'$.

## Proof

### $\epsilon$-$\delta$ Definition iff Definition by Limits

This is proved in Metric Space Continuity by Epsilon-Delta.

$\Box$

### $\epsilon$-$\delta$ Definition iff $\epsilon$-Ball Definition

This is proved in Metric Space Continuity by Open Ball.

$\blacksquare$

### $\epsilon$-Ball Definition iff Definition by Neighborhood

This is proved in Metric Space Continuity by Neighborhood.

$\blacksquare$