Tõkestamata intervallis ühtlaselt pidevad funktsioonid

Abstract

Uniform continuity of real functions on unbounded intervals is not a compulsory topic in the basic analysis courses. In the course «Calculus III», the study of the uniform continuity is limited to Cantor’s theorem. The theorem states that if f : [a; b] ! R is a continuous function, then it is uniformly continuous. This result is essential for proving one of the central results of the integral calculus, which states that if a function f : [a; b] ! R is continuous, then it is integrable on [a; b]. We also know that if f : [a;1) ! R is continuous and limx!1 f(x) exists (as a real number), then f is uniformly continuous on the interval [a;1). The aim of this bachelor thesis is to characterize functions f : [a;1) ! R that are uniformly continuous on [a;1). This thesis consists of three chapters and is based on the articles by R. L. Pouso [4] and S. Djebali [1]. In the first chapter of this bachelor thesis, we discuss the definition of a uniform continuity and bring some examples of continuous functions that are not uniformly continuous. In this introductory chapter we use sequences to investigate uniform continuity.We provide an important proposition on the relation between uniformly continuous function and Cauchy sequences, which we will use for further proofs. The second chapter is devoted to examining uniformly continuous functions on unbounded intervals. We present conditions for a function f : [a;1) ! R to be uniformly continuious on an interval [a;1). Then we give an example that shows that the condition limx!1 jf(x)j x = 0 does not guarantee uniform continuity of the function f. At the end of the chapter, we show that if the graph of a function f has a horizontal or oblique asymptote then the function is uniformly continuous. In the third chapter, the relation between uniform continuity and convergence of improper integrals is studied.We examine unifom continuity of a function f : [a;1) ! R, for which improper integral R 1 a f(x)dx convergences. We also provide example of cases when uniform continuity of a function does not guarantee integral convergence.