Definition:Continuous Mapping (Metric Space)/Point/Definition 1

Definition

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ 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 $A_1$.

$f$ 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.

Also known as

A mapping which is continuous at $a$ with respect to $d_1$ and $d_2$ can also be referred to as $\tuple {d_1, d_2}$-continuous at $a$.

Also see

• Equivalence of Definitions of Metric Space Continuity at Point

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis: $4.5$
• 1967: George McCarty: Topology: An Introduction with Application to Topological Groups ... (previous) ... (next): Chapter $\text{III}$: Metric Spaces: The Definition
• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 3$: Continuity: Definition $3.2$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $2$: Continuity generalized: metric spaces: $2.1$: Motivation: Definition $2.1.3$