Chapter I. Theory of Calculus in One Real Variable

Abstract

This chapter, beginning with Section 2, develops the topic of sequences and series of functions, especially of functions of one variable. An important part of the treatment is an introduction to the problem of interchange of limits, both theoretically and practically. This problem plays a role repeatedly in real analysis, but its visibility decreases as more and more results are developed for handling it in various situations. Fourier series are introduced in this chapter and are carried along throughout the book as a motivating example for a number of problems in real analysis.

Section 1 makes contact with the core of a first undergraduate course in real-variable theory. Some material from such a course is repeated here in order to establish notation and a point of view. Omitted material is summarized at the end of the section, and some of it is discussed in a little more detail in Appendix A at the end of the book. The point of view being established is the use of defining properties of the real number system to prove the Bolzano–Weierstrass Theorem, followed by the use of that theorem to prove some of the difficult theorems that are usually assumed in a one-variable calculus course. The treatment makes use of the extended real-number system, in order to allow sup and inf to be defined for any nonempty set of reals and to allow lim sup and lim inf to be meaningful for any sequence.

Sections 2–3 introduce the problem of interchange of limits. They show how certain concrete problems can be viewed in this way, and they give a way of thinking about all such interchanges in a common framework. A positive result affirms such an interchange under suitable hypotheses of monotonicity. This positive result is by way of introduction to the topic in Section 3 of uniform convergence and the role of uniform convergence in continuity and differentiation.

Section 4 gives a careful development of the Riemann integral for real-valued functions of one variable, establishing existence of Riemann integrals for bounded functions that are discontinuous at only finitely many points, basic properties of the integral, the Fundamental Theorem of Calculus for continuous integrands, the change-of-variables formula, and other results. Section 5 examines complex-valued functions, pointing out the extent to which the results for real-valued functions in the first four sections extend to complex-valued functions.

Section 6 is a short treatment of the version of Taylor's Theorem in which the remainder is given by an integral. Section 7 takes up power series and uses them to define the elementary transcendental functions and establish their properties. The power series expansion of $(1+x)^p$ for arbitrary complex $p$ is studied carefully. Section 8 introduces Cesàro and Abel summability, which play a role in the subject of Fourier series. A converse theorem to Abel's theorem is used to exhibit the function $|x|$ as the uniform limit of polynomials on $[-1,1]$. The Weierstrass Approximation Theorem of Section 9 generalizes this example and establishes that every continuous complex-valued function on a closed bounded interval is the uniform limit of polynomials.

Section 10 introduces Fourier series in one variable in the context of the Riemann integral. The main theorems of the section are a convergence result for continuously differentiable functions, Bessel's inequality, the Riemann–Lebesgue Lemma, Fejér's Theorem, and Parseval's Theorem.