Here we have a text that is very nice for the audience for which it was intended. It is meant as a last course in real analysis for those intending to go into teaching, giving them that necessary layer of knowledge above calculus so that they can teach calculus well and with confidence. At the author’s school, Appalachian State University, the course is called “Analysis for teachers”.
It is short, a mere 262 pages, but includes everything that should be included and then some: its historical material is valuable and included some items that were new to me. There are good exercises, though they appear only at the end of the five chapters, making the assignment of daily homework a little harder. I will refrain from listing its contents: a summary is here.
The writing is good, the author includes many references, the pictures are pretty, and the numerous italicized exhortations to students (e.g., “Verify this!”) may inspire some useful work.
This is an admirable text that I hope will be widely used.
Because we don’t want the author to get a swelled head, I will exercise my reviewer’s prerogative of pointing out unimportant mistakes: on page 119, “l’Hôpitals Rule” appears three times; to make up for the lack of apostrophes, “the Bernoulli’s” is on page 97. And poor L’Hôpital doesn’t get an entry in the index.
Look at this book! You’ll like it.
Woody Dudley learned his real analysis from Richard Goldberg’s text, published circa 1957. It was good too.
1 Elementary Calculus.
1.1 Preliminary Concepts.
1.2 Limits and Continuity.
1.5 Sequences and Series of Constants.
1.6 Power Series and Taylor Series.
Interlude: Fermat, Descartes, and theTangent Problem.
2 Introduction to Real Analysis.
2.1 Basic Topology of the Real Numbers.
2.2 Limits and Continuity.
2.4 Riemann and Riemann-Stieltjes Integration.
2.5 Sequences, Series, and Convergence Tests.
2.6 Pointwise and Uniform Convergence.
Interlude: Euler and the "Basel Problem".
3 A Brief Introduction to Lebesgue Theory.
3.1 Lebesgue Measure and Measurable Sets.
3.2 The Lebesgue Integral.
3.3 Measure, Integral, and Convergence.
3.4 Littlewood’s Three Principles.
Interlude: The Set of Rational Numbers isVery Large andVery Small.
4 Special Topics.
4.1 Modeling with Logistic Functions—Numerical Derivatives.
4.2 Numerical Quadrature.
4.3 Fourier Series.
4.4 Special Functions—The Gamma Function.
4.5 Calculus Without Limits: Differential Algebra.
Appendix A: Definitions and Theorems of Elementary Real Analysis.
A.3 The Derivative.
A.4 Riemann Integration.
A.5 Riemann-Stieltjes Integration.
A.6 Sequences and Series of Constants.
A.7 Sequences and Series of Functions.
Appendix B: A Very Brief Calculus Chronology.
Appendix C: Projects in Real Analysis.
C.1 Historical Writing Projects.
C.2 Induction Proofs: Summations, Inequalities, and Divisibility.
C.3 Series Rearrangements.
C.4 Newton and the Binomial Theorem.
C.5 Symmetric Sums of Logarithms.
C.6 Logical Equivalence: Completeness of the Real Numbers.
C.7 Vitali’s Nonmeasurable Set.
C.8 Sources for Real Analysis Projects.
C.9 Sources for Projects for Calculus Students.