Uniform Continuity: Definitions, Criteria, Examples, and Consequences

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Uniform continuity

For every ε > 0, one δ > 0 works for all domain pairs: if $d_1(x,y)<\delta$, then $d_2(f(x),f(y))<\varepsilon$.

Sequential criterion for uniform continuity

Whenever $d_1(x_n,y_n)\to0$, uniform continuity requires $d_2(f(x_n),f(y_n))\to0$.

Lipschitz function

A function satisfying $d_2(f(x),f(y))\le Ld_1(x,y)$ for some fixed $L\ge0$ and all domain pairs.

Heine–Cantor theorem

Every continuous function from a compact metric space to a metric space is uniformly continuous.

Sequential disproof of uniform continuity

Find input pairs whose distances tend to zero while their output distances do not.

Quiz questions

1. Which statement correctly describes the relationship between uniform continuity and continuity?Answer: Uniform continuity implies continuity, but continuity need not imply uniform continuity.
2. What follows directly from a Lipschitz estimate $d_2(f(x),f(y))\le Ld_1(x,y)$ with fixed $L\ge0$?Answer: The function is uniformly continuous.
3. What does the Heine–Cantor theorem guarantee for a continuous function on a compact metric space?Answer: The function is uniformly continuous.
4. For $f(x)=1/x$ on $(0,1)$, what do $x_n=1/n$ and $y_n=1/(n+1)$ show?Answer: Input distances tend to zero while output distances stay equal to 1.
5. What does uniform continuity guarantee about the image of a Cauchy sequence?Answer: The image sequence is Cauchy.