Uniform continuity : Uniformly continuous function
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 (xn) is a Cauchy sequence and f is a uniformly continuous function, then (f(xn)) 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.
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GLOSSARY
Aiwa, effendi----Yea, noble sir.
Allah-haly 'm alla-haly----A singsong of river-workers.
Allshu Akbar----God is most Great.
Antichi----Antiquities.
Backsheesh----Tip, douceur, bribe.
Basha----Pasha.
Bimbashi----Major.
Bismillah----In the name of God.
Darabukkeh----A drum made of a skin stretched over an earthenware funnel.
Holy Carpet from Mecca.
El aadah----The ordinary.
Fantasia----Celebration with music, dancing, and processions.
Fatihah----The opening chapter of the Koran, recited at weddings, etc.
Feddan----The most common measure of land--a little less than an acre.
Fellah (plu. fellaheen)----The Egyptian peasant.
Fessikh----Salted fish.
Ghawdzee----The tribe.html">tribe of public dancing-girls. A female.html">female of this tribe is
Ghawazee is generally understood as applying to the female.
Goolah----Porous water-jar of Nile mud.
Hanouti----Funeral attendants.
Hashish----Leaves of hemp.
Jibbeh----Long coat or smock, worn by dervishes.
Kemengeh----A cocoanut fiddle.
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