Definition:Uniform Continuity/Metric Space

Definition

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

Then a mapping $f: A_1 \to A_2$ is uniformly continuous on $A_1$ if and only if:

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

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

Also see

• Definition:Continuous Mapping (Metric Space)

• Definition:Absolute Continuity

• Results about uniformly continuous mappings in the context of metric spaces can be found here.

Sources

• 1953: Walter Rudin: Principles of Mathematical Analysis ... (previous) ... (next): $4.18$
• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $5$: Compact spaces: $5.8$: Compactness and Uniform Continuity: Definition $5.8.1$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): uniformly continuous function
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): uniformly continuous function
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): uniformly continuous function