# Uniform continuity

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A property of a function (mapping) $f: X \rightarrow Y$, where $X$ and $Y$ are metric spaces. It requires that for any $\epsilon > 0$ there is a $\delta > 0$ such that for all $x _ {1} , x _ {2} \in X$ satisfying $\rho ( x _ {1} , x _ {2} ) < \delta$, the inequality $\rho ( f ( x _ {1} ), f ( x _ {2} )) < \epsilon$ holds.

If a mapping $f: X \rightarrow Y$ is continuous on $X$ and $X$ is a compactum, then $f$ is uniformly continuous on $X$. The composite of uniformly-continuous mappings is uniformly continuous.

Uniform continuity of mappings occurs also in the theory of topological groups. For example, a mapping $f: X _ {0} \rightarrow Y$, where $X _ {0} \subset X$, $X$ and $Y$ topological groups, is said to be uniformly continuous if for any neighbourhood of the identity $U _ {y}$ in $Y$, there is a neighbourhood of the identity $U _ {x}$ in $X$ such that for any $x _ {1} , x _ {2} \in X _ {0}$ satisfying $x _ {1} x _ {2} ^ {-} 1 \in U _ {x}$( respectively, $x _ {1} ^ {-} 1 x _ {2} \in U _ {x}$), the inclusion $f ( x _ {1} ) [ f ( x _ {2} )] ^ {-} 1 \in U _ {y}$( respectively, $[ f ( x _ {1} )] ^ {-} 1 f ( x _ {2} ) \in U _ {y}$) holds.

The notion of uniform continuity has been generalized to mappings of uniform spaces (cf. Uniform space).

#### References

 [1] A.N. Kolmogorov, S.V. Fomin, "Elements of the theory of functions and functional analysis" , 1–2 , Graylock (1957–1961) (Translated from Russian) [2] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) [3] J.L. Kelley, "General topology" , Springer (1975) [4] N. Bourbaki, "General topology" , Elements of mathematics , Springer (1989) (Translated from French)