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 UniformlyContinuous 2013-03-22 12:45:38 2013-03-22 12:45:38 n3o (216) n3o (216) 14 n3o (216) Definition msc 26A15 UniformContinuity uniformly continuous function