Uniform continuity on a set

Definition

Let $D \subseteq \mathbb{R}$ , let $f \colon D \to \mathbb{R}$ be a function, and let $S \subseteq D$ . The function $f$ is uniformly continuous on $S$ if the following holds. For each $\varepsilon > 0$ there is $\delta > 0$ with $|f(x) - f(y)| < \varepsilon$ for all $x, y \in S$ with $|x - y| < \delta$ .

Remarks

So $f$ is uniformly continuous on $S$ when its restriction to $S$ is uniformly continuous. A function can be uniformly continuous on a set but not on a larger set. For example, $x^2$ is uniformly continuous on $[0, 1]$ but not on $\mathbb{R}$ . Mathlib defines it as UniformContinuousOn.

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Limit point of a set definition
Limit of a function definition
Continuity at a point definition
Epsilon-delta definition of continuity statement
Uniform continuity definition
Uniform continuity on a set definition

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Tag	`0A3CK6DVVB`
URI	`m2://0A3CK6DVVB`
Kind	definition
Writing	W0 stub
MSC	26A15
Defines	Uniform continuity on a set
Uses	Uniform continuity

Uniform continuity on a set

Definition

Remarks

Reading path

Where this leads

Further reading

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