Uniform continuity on unbounded intervals

Abstract

We present a teaching approach to uniform continuity on unbounded intervals which, hopefully, may help to meet the following pedagogical objectives: i. To provide students with efficient and simple criteria to decide whether a continuous function is also uniformly continuous; ii. To provide students with skill to recognize graphically significant classes of both uniformly and nonuniformly continuous functions. Assembling some well-known facts and refining the resulting statement, we establish a useful asymptotic coincidence test for the uniform continuity on unbounded intervals. That test is the core of the present note and yields an easily applicable technique. In particular, one of its immediate consequences is the elementary fact that continuity and existence of horizontal or oblique asymptotes imply uniform continuity.