Uniform Continuity

Uniform continuity is a property of a function that strengthens ordinary continuity by requiring the same control of output differences throughout its domain. For any prescribed positive tolerance in the output, there must be a positive tolerance in the input that works for every pair of domain points. Unlike ordinary continuity, this input tolerance cannot depend on where the points lie. The definition applies to functions between metric spaces. (jirka.org)

Definition and distinction from continuity

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A function $f:X\to Y$ is uniformly continuous if

For a real-valued function on a subset of the real numbers, this becomes

The number $\delta$ may depend on $\varepsilon$, the function, and the chosen domain and metrics, but not on $x$ or $y$. (jirka.org)

For a continuous function, the corresponding quantifier order is

Here $\delta$ may depend on $x$. Uniform continuity therefore implies continuity, but continuity need not imply uniform continuity. The distinction concerns uniform control over a whole domain, not merely behavior near each individual point. (jirilebl.github.io)

Negating the definition gives a useful test: uniform continuity fails precisely when there is some $\varepsilon_0>0$ such that, for every $\delta>0$, two domain points can be found with input distance less than $\delta$ but output distance at least $\varepsilon_0$. (jirilebl.github.io)

Examples and dependence on the domain

The following examples illustrate different mechanisms; the displayed estimates directly verify the definition or its negation. (jirilebl.github.io)

• Linear functions. If $f(x)=ax+b$, then

  $$|f(x)-f(y)|=|a|\,|x-y|.$$

  For $a\ne0$, choosing $\delta=\varepsilon/|a|$ proves uniform continuity on all of $\mathbb R$. Constant functions are uniformly continuous as well.

• The square function. The function $f(x)=x^2$ is continuous but not uniformly continuous on $\mathbb R$. Set

  $$x_n=n,\qquad y_n=n+\frac1n.$$

  Then $|x_n-y_n|=1/n\to0$, whereas

  $$|y_n^2-x_n^2|=2+\frac1{n^2}.$$

  On any bounded interval, however, the factorization

  $$|x^2-y^2|=|x-y|\,|x+y|$$

  supplies a uniform bound.

• Reciprocal functions near a missing endpoint. On $(0,1)$, $f(x)=1/x$ is continuous but not uniformly continuous. Taking $x_n=1/n$ and $y_n=1/(n+1)$, for $n\ge2$, gives input distances tending to zero while the output difference remains $1$.

• The square-root function. On $[0,\infty)$,

  $$|\sqrt{x}-\sqrt{y}|\le\sqrt{|x-y|}.$$

  Thus $\delta=\varepsilon^2$ works everywhere. Uniform continuity does not require a bounded domain, a bounded function, or a bounded derivative.

Restricting a uniformly continuous function to a smaller domain preserves uniform continuity. Enlarging its domain may destroy it, as the square-function example shows. (jirka.org)

Compactness and the Heine–Cantor theorem

The Heine–Cantor theorem states that every continuous function from a compact metric space into a metric space is uniformly continuous. In particular, every continuous real-valued function on a closed bounded interval $[a,b]$ is uniformly continuous. (jirka.org)

A sequential proof explains the role of compactness. If uniform continuity failed, there would be sequences $x_n,y_n$ satisfying

Compactness supplies a subsequence $x_{n_k}\to x$. The corresponding $y_{n_k}$ also tends to $x$. Continuity forces both image subsequences to tend to $f(x)$, contradicting their fixed separation. (jirilebl.github.io)

Compactness is sufficient, not necessary: linear functions are uniformly continuous on the noncompact domain $\mathbb R$. Conversely, boundedness of a domain alone is insufficient, as $1/x$ on $(0,1)$ demonstrates. (jirka.org)

Quantitative sufficient conditions

Lipschitz continuity provides a stronger, quantitative condition. If a constant $L\ge0$ satisfies

for all $x,y$, then $f$ is uniformly continuous. When $L>0$, one may take $\delta=\varepsilon/L$. (jirilebl.github.io)

For a differentiable real function on an interval, a uniform bound $|f'(x)|\le M$ implies a Lipschitz estimate through the mean value theorem. This is a sufficient condition, not a necessary one: the square-root estimate above establishes uniform continuity even though its derivative becomes unbounded near zero. (jirka.org)

More generally, a global estimate

called Hölder continuity, proves uniform continuity by the choice

This illustrates how an explicit rate of output variation can strengthen the qualitative epsilon–delta requirement. (math.rice.edu)

Cauchy sequences and extension

Uniformly continuous functions preserve Cauchy sequences. Given an output tolerance $\varepsilon$, choose its uniform input tolerance $\delta$. Eventually all pairs of terms of a Cauchy sequence lie within $\delta$, so all corresponding image terms lie within $\varepsilon$. (jirka.org)

This leads to an extension theorem. If $A$ is a dense subset of a metric space $X$, $Y$ is a complete metric space, and $f:A\to Y$ is uniformly continuous, then $f$ has a unique uniformly continuous extension $F:X\to Y$. For $a_n\in A$ with $a_n\to x$, define

Completeness ensures existence of the limit, and uniform continuity ensures that it is independent of the approximating sequence. (math.uwaterloo.ca)

In particular, a uniformly continuous map into a complete metric space extends uniquely to the completion of its domain. For finite real endpoints $a<b$, a function $(a,b)\to\mathbb R$ is uniformly continuous exactly when it admits a continuous extension to $[a,b]$. (leanprover-community.github.io)

Limits and integration

Uniform continuity is distinct from uniform convergence: the former compares values of one function at different inputs, while the latter compares different functions at the same input. Nevertheless, a uniform limit of uniformly continuous functions is uniformly continuous. (leanprover-community.github.io)

The proof uses the triangle inequality:

Uniform convergence makes the first and third terms uniformly small; uniform continuity of one suitable $f_n$ controls the middle term. No common choice of $\delta$ for all $f_n$ is required. (leanprover-community.github.io)

Uniform continuity also underlies the proof that a continuous function on $[a,b]$ has a Riemann integral. A sufficiently fine partition makes the function’s oscillation small on every subinterval simultaneously, so upper and lower sums differ by arbitrarily little. (ocw.mit.edu)

Metric dependence and uniform spaces

Ordinary continuity depends only on the topology induced by the metrics. Uniform continuity can distinguish metrics that induce the same topology. On $\mathbb R$, for example, the metrics

give the same topology. The identity map from $(\mathbb R,d)$ to $(\mathbb R,\rho)$ is uniformly continuous, but the reverse map is not: $n$ and $n+1$ become arbitrarily close in $\rho$ while remaining distance $1$ apart in $d$. (homepages.ecs.vuw.ac.nz)

The broader setting is a uniform space, which specifies a uniform notion of closeness without necessarily using a metric. Uniform continuity then means that every prescribed target closeness is guaranteed by some source closeness, uniformly for all pairs of points. This framework retains the connection between uniform continuity, Cauchy behavior, and completion. (leanprover-community.github.io)

References

1. RA Continuous functionsjirka.org
2. RA Uniform continuityjirilebl.github.io
3. Basic Analysis Ijirka.org
4. MIT18_100af20_lec17.pdfocw.mit.edu
5. Kenneth R. Davidson, Allan P. Donsig: Real Analysis and Applications supplementary materialmath.uwaterloo.ca
6. Extension by continuitywww-users.cse.umn.edu
7. Mathlib.Topology.UniformSpace.Completionleanprover-community.github.io
8. Mathlib.Topology.UniformSpace.UniformApproximationleanprover-community.github.io
9. mit18_100af20_lec21.pdfocw.mit.edu
10. MATH 452 General Topologyhomepages.ecs.vuw.ac.nz
11. Some topics in analysis related to metricsmath.rice.edu