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# 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).

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## Comments

## Metric spaces

Maybe we could mention that the same definition can be used for any function between two metric spaces (or even uniform spaces) and that uniformly continuous functions map Cauchy sequences to Cauchy sequences and are therefore continuous.