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).