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"). Analysis is the branch of mathematics most explicitly concerned with the notion of a limit, either the limit of a sequence or the limit of a function. ... Partial plot of a function f. ...

Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and when we speak of a function being continuous on an interval, we mean only that it is continuous at each point of the interval. In contrast, uniform continuity is a global property of a function. Uniform continuity may also be defined for an interval. Any function continious on a closed interval is also uniformly continious on that interval. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Definition

Given metric spaces (X,d₁) and (Y,d₂), if and then a function is called uniformly continuous if for every real number ε > 0 there exists δ > 0 such that for all with d₁(x,y) < δ, we have that d₂(f(x),f(y)) < ε. In mathematics, a metric space is a set (or space) where a distance between points is defined. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

If X and Y are subsets of the real numbers, d₁ and d₂ can be the standard Euclidian norm, , yielding the definition: if for all ε > 0 there exists a δ > 0 such that | x − y | < δ implies | f(x) − f(y) | < ε. In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ...

Properties

Every uniformly continuous function is continuous, but the converse is not 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. In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ...

If M is a compact metric space, then every continuous f : M → N is uniformly continuous (this is the Heine-Cantor theorem). In particular, if a function is continuous at every point of a closed bounded interval, it is uniformly continuous on that interval. In mathematics, a subset of Euclidean space Rn is called compact if it is closed and bounded. ... In mathematics, the Heine-Cantor theorem states that if M is a compact metric space, then every continuous function f : M &#8594; N where N is a metric space is uniformly continuous. ... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...

Every Lipschitz continuous map between two metric spaces is uniformly continuous. In mathematics, a function f : M &#8594; N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) &#8804; K d(x, y) for all x and y...

If (x_n) is a Cauchy sequence contained in the domain of f (though perhaps not convergent in the domain of f) and f is a uniformly continuous function, then (f(x_n)) is also a Cauchy sequence. In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses. ...

If a function f is uniformly continuous over a finite interval (a,b), then f is also bounded over (a,b).

Generalization to topological vector spaces

In the special case of two topological vector spaces V and W, the notion of uniform continuity of a map becomes : for any neighborhood B of zero in W, there exists a neighborhood A of zero in V such that implies . In mathematics, a topological vector space X is a real or complex vector space which is endowed with a Hausdorff topology such that vector addition X × X &#8594; X and scalar multiplication K × X &#8594; X are continuous (where the product topologies are used and the base field K carries...

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 the mathematical field of topology, a uniform space is a set with a uniform structure. ...

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.