Uniform continuity

Builds on Epsilon-delta definition of continuity

Definition

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

Remarks

Each uniformly continuous function is continuous. In the epsilon-delta form of continuity, $\delta$ may depend on the point. Here one $\delta$ works for all points. The function $x^2$ is continuous on $\mathbb{R}$ but not uniformly continuous. Mathlib defines it for uniform spaces, as UniformContinuous. Eduard Heine published the first definition of uniform continuity, in 1870.

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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

Where this leads

25 pages build on this one.

Tag	`0Q4DR37DVE`
URI	`m2://0Q4DR37DVE`
Kind	definition
Writing	W1 stated
MSC	26A15
Defines	Uniform continuity
Uses	Epsilon-delta definition of continuity

Uniform continuity

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W2

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