uniformly continuous

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

\forall\varepsilon>0\ \exists\delta>0\ \forall x,y\in A\ |x-y|<\delta% \Rightarrow|f(x)-f(y)|<\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 $f(x)=1/x$ 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 $X$ and $Y$ are metric spaces with distances $d_{X}$ and $d_{Y}$ , we say that $f$ is uniformly continuous if

\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).

Title	uniformly continuous
Canonical name	UniformlyContinuous
Date of creation	2013-03-22 12:45:38
Last modified on	2013-03-22 12:45:38
Owner	n3o (216)
Last modified by	n3o (216)
Numerical id	14
Author	n3o (216)
Entry type	Definition
Classification	msc 26A15
Related topic	UniformContinuity
Defines	uniformly continuous function