Uniform continuity

Let (X,d_X) and (Y,d_Y) be metric spaces. A function f: X → Y is uniformly continuous if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ X, if d_X(x, y) < δ then d_Y(f(x), f(y)) < ε.¹ The key distinction from ordinary continuity is that δ depends only on ε, not on the point x. For example, f(x) = x² on ℝ is continuous but not uniformly continuous, because as x grows, a fixed δ produces larger changes in f(x). In contrast, f(x) = sin(x) is uniformly continuous on ℝ because its derivative is bounded.

The Heine–Cantor theorem states that every continuous function on a compact metric space is uniformly continuous.² This result underpins many applications in analysis, such as the existence of the Riemann integral for continuous functions on closed intervals. Another fundamental property is that a uniformly continuous function maps Cauchy sequences to Cauchy sequences, a fact that is not true for merely continuous functions.³ This allows the extension of uniformly continuous functions from dense subsets to their closures, a technique used in constructing completions of metric spaces and in defining the integral of continuous functions on abstract spaces.

Uniform continuity sits between ordinary continuity and stronger conditions such as Lipschitz continuity and Hölder continuity. A Lipschitz function satisfies |f(x) − f(y)| ≤ K|x − y|, which immediately implies uniform continuity with δ = ε/K. However, the converse fails: the function f(x) = √x on [0,1] is uniformly continuous but not Lipschitz, because its derivative is unbounded near 0.⁴ Hölder continuity, where |f(x) − f(y)| ≤ C|x − y|^α for 0 < α ≤ 1, also implies uniform continuity. In the broader setting of uniform spaces, uniform continuity is defined via entourages, generalizing the metric definition to topological groups and uniform spaces.

Uniform continuity has surprising connections to number theory and dynamical systems. For instance, the function f(x) = sin(1/x) is continuous on (0,1) but not uniformly continuous, yet its restriction to any compact subinterval is uniformly continuous. A less-known result is that a function on a bounded interval is uniformly continuous if and only if it has a continuous extension to the closure of the interval.⁵ In the theory of topological groups, a uniformly continuous function on a dense subgroup extends uniquely to the whole group, a property used in the construction of the Bohr compactification. Additionally, the concept of uniform continuity is essential in the definition of the uniform structure of a metric space, which underlies the notion of completeness and total boundedness.⁶ A niche application appears in computer science: in the theory of computable analysis, uniformly continuous functions on compact domains are exactly those that can be effectively approximated by algorithms.⁷

Metric space

A set equipped with a distance function satisfying the axioms of a metric.

Compact space

A topological space in which every open cover has a finite subcover.

Cauchy sequence

A sequence whose elements become arbitrarily close to each other as the sequence progresses.

Lipschitz continuity

A strong form of uniform continuity where the ratio of output change to input change is bounded by a constant.

Hölder continuity

A generalization of Lipschitz continuity with exponent α between 0 and 1.

Uniform space

A set with a uniform structure that generalizes the notion of nearness in metric spaces.

Bohr compactification

A compact topological group associated with a given topological group, used in harmonic analysis.