General properties of real-valued functions

Abstract

Many general properties of real continuous functions defined on the closed interval [0,1] on the real line have been studied in the past. The present work started with a review of some known analytical results of this class C. An extended class A of positively continuous functions was then defined and its properties compared with that of C. While many of the characteristics of C were inherited from A, some properties of C are not shared by A.

The first part of the second section (chapter) is devoted to a study of some elementary algebraic properties of A and C. Results obtained for the two classes showed differences. The rest of the section (chapter) deals with the algebraic structures of some subclasses of F, the set of all real-valued functions of [0,1] on the real line. The concept of an ideal in F was introduced for the class of real functions from [0,1] to the real line.

In the last section (chapter), the concept of areas of function, C_f, defined as the closure of the graph of a function, is used to study the properties of elements of I. Integrals of continuous functions in I are completely determined by their C_f’s. Some topological implications of a few analytical subclasses of I were also revealed. This section concluded with an important theorem that fully characterizes the G_f of a real function in I by a closed set in the closed square U = {(x,y) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1}.