Uniform Continuity: Another Way to Approach This Concept in the Classroom

Abstract

Let f be a continuous function defined on an interval J . We say that a function δ from J × R+ to R+ is a delta-epsilon function for f on J , if for all p ∈ J and > 0, the number δ(p, ) satisfies the epsilon-delta definition of continuity of f at p for that . More precisely, δ(p, ) is such that for all x ∈ J with |x − p| < δ(p, ), then |f (x) − f (p)| < . Next, we provide necessary and sufficient conditions to analyze the uniform continuity of the function f based on the behavior of the function η : R+ → [0, ∞) given by