Uniform Continuity

analysis

Definition

Uniform Continuity

A function is uniformly continuous on if every requested output tolerance can be met by one input tolerance that works everywhere in the domain. Formally, for every , there exists such that, for all ,

The essential point is that may depend on , but it may not depend on where and lie in . One horizontal scale must control the whole domain.

Difference from pointwise continuity

Ordinary continuity is local. At each point , and for each , one may choose a radius that depends on both and .

Uniform continuity removes that dependence on . After is fixed, the same must work for every location in the domain. This prevents the function from becoming more and more sensitive as one moves through the domain.

Symbolically, pointwise continuity allows

while uniform continuity requires

Why compact domains help

The Heine-Cantor theorem states that every continuous real-valued function on a compact domain is uniformly continuous. Compactness prevents the domain from hiding infinitely many places where the required becomes smaller and smaller without limit.

This is why a function can be continuous but fail to be uniformly continuous on an open or unbounded domain. For example, is continuous on , but it is not uniformly continuous there because the function becomes arbitrarily steep near .

Relation to Lipschitz continuity

Every Lipschitz continuous function is uniformly continuous. Lipschitz continuity gives the explicit bound

so one can choose when .

The converse is false. Uniform continuity controls the same input tolerance across the domain, but it does not require a fixed linear bound on all secant slopes. The function on is uniformly continuous, but not Lipschitz continuous.