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A property of a function (mapping) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952201.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952202.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952203.png" /> are metric spaces. It requires that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952204.png" /> there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952205.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952206.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952207.png" />, the inequality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952208.png" /> holds.
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If a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u0952209.png" /> is continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522011.png" /> is a compactum, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522012.png" /> is uniformly continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522013.png" />. The composite of uniformly-continuous mappings is uniformly continuous.
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Uniform continuity of mappings occurs also in the theory of topological groups. For example, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522014.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522017.png" /> topological groups, is said to be uniformly continuous if for any neighbourhood of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522018.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522019.png" />, there is a neighbourhood of the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522020.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522021.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522022.png" /> satisfying <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522023.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522024.png" />), the inclusion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522025.png" /> (respectively, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/u/u095/u095220/u09522026.png" />) holds.
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A property of a function (mapping)  $  f:  X \rightarrow Y $,
 +
where  $  X $
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and  $  Y $
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are metric spaces. It requires that for any  $  \epsilon > 0 $
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there is a  $  \delta > 0 $
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such that for all  $  x _ {1} , x _ {2} \in X $
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satisfying  $  \rho ( x _ {1} , x _ {2} ) < \delta $,
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the inequality  $  \rho ( f ( x _ {1} ), f ( x _ {2} )) < \epsilon $
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holds.
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If a mapping  $  f:  X \rightarrow Y $
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is continuous on  $  X $
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and  $  X $
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is a compactum, then  $  f $
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is uniformly continuous on  $  X $.
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The composite of uniformly-continuous mappings is uniformly continuous.
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Uniform continuity of mappings occurs also in the theory of topological groups. For example, a mapping $  f:  X _ {0} \rightarrow Y $,  
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where $  X _ {0} \subset  X $,  
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$  X $
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and $  Y $
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topological groups, is said to be uniformly continuous if for any neighbourhood of the identity $  U _ {y} $
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in $  Y $,  
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there is a neighbourhood of the identity $  U _ {x} $
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in $  X $
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such that for any $  x _ {1} , x _ {2} \in X _ {0} $
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satisfying $  x _ {1} x _ {2}  ^ {-} 1 \in U _ {x} $(
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respectively, $  x _ {1}  ^ {-} 1 x _ {2} \in U _ {x} $),  
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the inclusion $  f ( x _ {1} ) [ f ( x _ {2} )]  ^ {-} 1 \in U _ {y} $(
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respectively, $  [ f ( x _ {1} )]  ^ {-} 1 f ( x _ {2} ) \in U _ {y} $)  
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holds.
  
 
The notion of uniform continuity has been generalized to mappings of uniform spaces (cf. [[Uniform space|Uniform space]]).
 
The notion of uniform continuity has been generalized to mappings of uniform spaces (cf. [[Uniform space|Uniform space]]).
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====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,  "General topology" , ''Elements of mathematics'' , Springer  (1989)  (Translated from French)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.N. Kolmogorov,  S.V. Fomin,  "Elements of the theory of functions and functional analysis" , '''1–2''' , Graylock  (1957–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  J.L. Kelley,  "General topology" , Springer  (1975)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N. Bourbaki,  "General topology" , ''Elements of mathematics'' , Springer  (1989)  (Translated from French)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====

Latest revision as of 08:27, 6 June 2020


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)

Comments

There are several natural uniform structures on a topological group; the (confusing) statement above about uniform continuity of mappings between them can be interpreted in various ways.

References

[a1] W. Roelcke, S. Dierolf, "Uniform structures on topological groups and their quotients" , McGraw-Hill (1981)
[a2] R. Engelking, "General topology" , Heldermann (1989)
How to Cite This Entry:
Uniform continuity. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Uniform_continuity&oldid=12797
This article was adapted from an original article by L.D. Kudryavtsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article