## Uniform Continuity

The ordinarydefinition of continuity states:

A functionis continuous at a pointif forgiven there existssuch thatIn this definitionis generally a function ofOften the function needs to be defined to an interval, so thatis restricted to a certain range, for example

If a function is uniformly continuous then the same value ofcan be used for allgiven the sameUniform continuity is a stricter definition of continuity than thecriterion.

Definition A functionis uniformly continuous onif for everythere issuch that ifwiththenIfis uniformly continuous onwe say thatis uniformly continuous.

Example:with

Sinceso

Ifthenso takeandis fixed for fixed

Example:with

Sinceso

Ifthenso takeandis fixed for fixed