The Theory of Functions of a Real Variable

Abstract

Introduction. In this article, the theory of functions of a real variable will be considered from the narrowest possible point of view, as a theory set up to examine critically the processes of analysis and to extend and generalize these processes to as many as possible of the situations that arise in special branches of analysis, such as potential theory, differential equations, and trigonometric series. Actually it is difficult to fix precisely the boundaries of the subject, since to a considerable extent it overlaps with topology, logic and foundations, and the theory of abstract spaces. University courses in the subject are likely to vary extremely widely from one to the other in the subject matter presented and the point of view of the presentation. However, the accomplishment of the aim set forth above will usually occupy a central position, be it ever so disguised. The theory of functions of a real variable is based on the theory of the real numbers; consequently, one of our first tasks is to look at the real numbers and determine just what properties they must be assumed to have in order to develop the theory. It will be assumed that the reader has an intuition for the rational numbers (integers and fractions; positive, negative, and zero) and is familiar with the rational operations of addition, multiplication, subtraction, and division, and the laws governing those. Also it is assumed that he is familiar with the notion of order (a less than b, written a < b) and the most common rules pertaining to it. In algebraic language, it is assumed that the reader knows that the rational numbers form an ordered Archimedian field. How this is known is a subject of its own and will not be gone into here. The existence of such a system may be postulated outright, or a system may be constructed by suitable definitions from the system of positive integers, whose existence and basic properties (very few in number) may be postulated outright, or various other approaches may be used. In this matter our field of study is closely connected with logic and foundations, and with algebra. At any rate, let us assume that the rational numbers and their arithmetic are at our disposal, but that our intuition fails to provide us with any further numbers. Our system, then, is not sufficiently general to satisfy many of the demands of the processes of analysis. For example, the equation x2 = 2 has no solution, and the expression (1 + 1/n)n, n = 1, 2, 3, ... does not have any limit as n increases, in spite of the fact that it always increases and remains less than 3. The question arises then: what further properties should the real numbers possess in order to eliminate the specific difficulties mentioned and a host of similar difficulties? It turns out that one property is needed; the real numbers should be complete, in the sense of