Uniform continuity : Uniformly continuous

In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) ("continuity"), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself ("uniformity").

Definition

The formal definition is as follows: a function f : M -> N between metric spaces is called uniformly continuous if for every real number ε > 0 there exists a number δ > 0 such that for all x₁, x₂ in M with d(x₁, x₂) < δ, we have d(f(x₁), f(x₂)) < ε.

Properties

Every uniformly continuous function is continuous, but the converse isn't true. Consider for instance the function f(x) = 1/x with domain the positive real numbers. This function is continuous, but not uniformly continuous, since as x approaches 0, the changes in f(x) grow beyond any bound.

If M is a compact metric space, then every continuous f : M -> N is uniformly continuous.

Every Lipschitz continuous map between two metric spaces is uniformly continuous.

If (x_n) is a Cauchy sequence and f is a uniformly continuous function, then (f(x_n)) is also a Cauchy sequence.

Generalization to uniform spaces

The most natural and general setting for the study of uniform continuity are the uniform spaces. A function f : X -> Y between uniform space is called uniformly continuous if for every entourage V in Y there exists an entourage U in X such that for every (x₁, x₂) in U we have (f(x₁), f(x₂)) in V.

In this setting, it is also true that uniformly continuous maps transform Cauchy sequences into Cauchy sequences and that continuous maps on compact uniform spaces are automatically uniformly continuous.

Common misspelling and questions (FAQ)

niformly-continuous  uiformly-continuous  unformly-continuous  uniormly-continuous  unifrmly-continuous  unifomly-continuous  uniforly-continuous  uniformy-continuous  uniforml-continuous  uniformlycontinuous  uniformly-ontinuous  uniformly-cntinuous  uniformly-cotinuous  uniformly-coninuous  uniformly-contnuous  uniformly-contiuous  uniformly-continous  uniformly-continuus  uniformly-continuos  uniformly-continuou  nuiformly-continuous  uinformly-continuous  unfiormly-continuous  uniofrmly-continuous  unifromly-continuous  unifomrly-continuous  uniforlmy-continuous  uniformyl-continuous  uniforml-ycontinuous  uniformlyc-ontinuous  uniformly-ocntinuous  uniformly-cnotinuous  uniformly-cotninuous  uniformly-conitnuous  uniformly-contniuous  uniformly-contiunous  uniformly-continouus  uniformly-continuuos  uniformly-continuosu  uniformly-continuou  uuniformly-continuous  unniformly-continuous  uniiformly-continuous  unifformly-continuous  unifoormly-continuous  uniforrmly-continuous  uniformmly-continuous  uniformlly-continuous  uniformlyy-continuous  uniformly--continuous  uniformly-ccontinuous  uniformly-coontinuous  uniformly-conntinuous  uniformly-conttinuous  uniformly-contiinuous  uniformly-continnuous  uniformly-continuuous  uniformly-continuoous  uniformly-continuouus  uniformly-continuouss  7niformly-continuous  yniformly-continuous  hniformly-continuous  8niformly-continuous  jniformly-continuous  8niformly-continuous  iniformly-continuous  jniformly-continuous  uhiformly-continuous  ubiformly-continuous  ujiformly-continuous  ujiformly-continuous  umiformly-continuous  un8formly-continuous  unuformly-continuous  unjformly-continuous  un9formly-continuous  unkformly-continuous  un9formly-continuous  unoformly-continuous  unkformly-continuous  unirormly-continuous  unidormly-continuous  unicormly-continuous  unitormly-continuous  univormly-continuous  unitormly-continuous  unigormly-continuous  univormly-continuous  unif9rmly-continuous  unifirmly-continuous  unifkrmly-continuous  unif0rmly-continuous  uniflrmly-continuous  unif0rmly-continuous  unifprmly-continuous  uniflrmly-continuous  unifo4mly-continuous  unifoemly-continuous  unifodmly-continuous  unifo5mly-continuous  unifofmly-continuous  unifo5mly-continuous  unifotmly-continuous  unifofmly-continuous  uniforjly-continuous  unifornly-continuous  uniforkly-continuous  uniforkly-continuous  unifor,ly-continuous  uniformoy-continuous  uniformky-continuous  uniform,y-continuous  uniformpy-continuous  uniform.y-continuous  uniformpy-continuous  uniform;y-continuous  uniform.y-continuous  uniforml6-continuous  uniformlt-continuous  uniformlg-continuous  uniforml7-continuous  uniformlh-continuous  uniforml7-continuous  uniformlu-continuous  uniformlh-continuous  uniformly0continuous  uniformlypcontinuous  uniformly[continuous  uniformly-dontinuous  uniformly-xontinuous  uniformly-fontinuous  uniformly-fontinuous  uniformly-vontinuous  uniformly-c9ntinuous  uniformly-cintinuous  uniformly-ckntinuous  uniformly-c0ntinuous  uniformly-clntinuous  uniformly-c0ntinuous  uniformly-cpntinuous  uniformly-clntinuous  uniformly-cohtinuous  uniformly-cobtinuous  uniformly-cojtinuous  uniformly-cojtinuous  uniformly-comtinuous  uniformly-con5inuous  uniformly-conrinuous  uniformly-confinuous  uniformly-con6inuous  uniformly-conginuous  uniformly-con6inuous  uniformly-conyinuous  uniformly-conginuous  uniformly-cont8nuous  uniformly-contunuous  uniformly-contjnuous  uniformly-cont9nuous  uniformly-contknuous  uniformly-cont9nuous  uniformly-contonuous  uniformly-contknuous  uniformly-contihuous  uniformly-contibuous  uniformly-contijuous  uniformly-contijuous  uniformly-contimuous  uniformly-contin7ous  uniformly-continyous  uniformly-continhous  uniformly-contin8ous  uniformly-continjous  uniformly-contin8ous  uniformly-continious  uniformly-continjous  uniformly-continu9us  uniformly-continuius  uniformly-continukus  uniformly-continu0us  uniformly-continulus  uniformly-continu0us  uniformly-continupus  uniformly-continulus  uniformly-continuo7s  uniformly-continuoys  uniformly-continuohs  uniformly-continuo8s  uniformly-continuojs  uniformly-continuo8s  uniformly-continuois  uniformly-continuojs  uniformly-continuouw  uniformly-continuoua  uniformly-continuouz  uniformly-continuoue  uniformly-continuoux  uniformly-continuoue  uniformly-continuoud  uniformly-continuoux  unyformly-continuous  uniformly-conyinuous  uniformly-contyinuous  uniformly-continuoues 


However, the opportunity of taking the revenge he deserved gone, she sat down to dress, and was attired by her women to the best finest and largest diamonds set in gold, her necklace of pearls, six on largest ever seen, and invaluable, that the greatest sultanesses would bracelets, which were of diamonds and rubies intermixed, corresponded consulted her glass and women upon her adjustment; and when she found magician, she sat down on a sofa expecting his arrival. The magician came at the usual hour, and as soon as he entered the enchanting grace and smile, and pointed with her hand to the most same time, which was a civility she had never shown him before. The African magician, dazzled more with the luster of the princess's much surprised. The smiling and graceful air with which she received broke silence first, looking at him all the time in such a manner as to to understand hitherto, and said, "You are doubtless amazed to find.