Uniform continuity and growth of real continuous functions

Abstract

The purpose of this paper is to understand whether there exists any link between the uniform continuity of a real function defined on an unbounded interval and its growth at infinity. The primary objective is to present some results from teaching experience which help in the comprehension of this notion and yield some classroom techniques. It is well known that a uniformly continuous function has a monomial growth; it will be proved that there does not exist another growth of positive order. After introducing three kinds of growth, some results are recalled in connection with the behaviour near infinity of a uniformly continuous function. Using a series of counterexamples, it is shown that the uniform continuity of a function cannot be described by its asymptotic behaviour near infinity. Finally, some useful properties of the averaging convergence are reviewed, and how this is related to uniform continuity is investigated.