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uniformly continuous

(Definition)

Let $f: A \rightarrow \mathbb{R}$ be a real function defined on a subset of the real line. We say that is uniformly continuous if, given an arbitrary small positive $\varepsilon$ , there exists a positive $\delta$ such that whenever two points in differ by less than $\delta$ , they are mapped by into points which differ by less than $\varepsilon$ . In symbols:

$\displaystyle \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in A\ \vert x-y\vert < \delta \Rightarrow \vert f(x)-f(y)\vert < \varepsilon.$

Every uniformly continuous function is also continuous, while the converse does not always hold. For instance, the function $f: ]0,+\infty[ \rightarrow \mathbb{R}$ defined by is continuous in its domain, but not uniformly.

A more general definition of uniform continuity applies to functions between metric spaces (there are even more general environments for uniformly continuous functions, i.e. uniform spaces). Given a function $f: X \rightarrow Y$ , where and are metric spaces with distances and , we say that is uniformly continuous if

$\displaystyle \forall \varepsilon > 0\ \exists \delta > 0\ \forall x,y \in X\ d_X(x,y) < \delta \Rightarrow d_Y(f(x),f(y)) < \varepsilon.$

Uniformly continuous functions have the property that they map Cauchy sequences to Cauchy sequences and that they preserve uniform convergence of sequences of functions.

Any continuous function defined on a compact space is uniformly continuous (see Heine-Cantor theorem).

"uniformly continuous" is owned by n3o.

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See Also: uniform continuity

Also defines:

uniformly continuous function

Attachments:: uniformly continuous on $\mathbb{R}$ is roughly linear (Theorem) by Mathprof

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Cross-references: Heine-Cantor theorem, compact, sequences, uniform convergence, preserve, Cauchy sequences, map, property, distances, uniform spaces, even, metric spaces, uniform continuity, domain, function, converse, continuous, points, positive, line, real, subset, real function
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This is version 11 of uniformly continuous, born on 2002-06-07, modified 2006-09-21.
Object id is 3068, canonical name is UniformlyContinuous.
Accessed 21266 times total.

Classification:

AMS MSC:

26A15 (Real functions :: Functions of one variable :: Continuity and related questions )

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