From Calculus to Analysis

This is a fairly conventional introductory real analysis text. The “From Calculus” in the title is misleading, because the book does not assume knowledge of any facts from calculus, and it is a full-fledged proofs course, not a transition course.

The first half of the book proves all the important theorems from differential and integral calculus. The next quarter of the book deals with sequences and series, and in particular with trigonometric functions and Fourier series. The last quarter of the book deals with related and supporting materials such as point-set topology, logic, set theory, and the field axioms.

The book includes a number of gems not usually found at this level, such as a proof of the fundamental theorem of algebra, a discussion of convex functions and Jensen’s inequality, a nowhere-differentiable function, and uniform distribution of sequences. There’s also a proof that \(e\) is transcendental, although this is so intricate I doubt any students at this level will make it through the proof.

One unusual feature of the book is the approach to defining the reals, which is done as infinite decimals rather than Cauchy sequences or Dedekind cuts. I’m not enthusiastic about this approach, because it gives an artificial emphasis to base 10. It also has some subtleties, such as the fact that two infinite decimals can represent the same number (\(0.999\dots = 1.000\dots\). This particular subtlety is glossed over in the present book, which says flatly that the two decimals are “equal”. But it is a workable approach and and handled reasonably well here.

The treatment of Riemann integrals is awkward. They are defined in terms of integrals of upper and lower step functions rather than in terms of upper and lower sums. This is OK, but awkward because of the peculiar way step functions are defined: they are always zero at the partition (transition) points, so there is always added language about inequalities holding except at these points.

This book somehow escaped the copyediting process. It is loaded with partial sentences and some misspelled words. Here’s the first sentence in Chapter 6 (p. 91): “Local properties algebra derivative and the relationship between the sign of the derivative at a point and the function being monotone at a point.” Here’s the caption on p. 111: “A convex curve \(y=f(x)\) and one a it’s cords” (the word chord is spelled cord throughout the book). A ballistics example on p. 10 is about canons. In most cases the meaning can be inferred easily, but it’s jarring to read such sentences.

The mathematical content is good, and I like seeing a good assortment of enrichment topics as this book provides. Apart from the copyediting, there are a very few sloppy places, such as in Example 10.1.11 on p. 208 where the book proves the convergence of an infinite sum by manipulating the infinite sum rather than a finite partial sum. The book uses reversed brackets for open intervals, thus \(]0,1[\), which is very uncommon but should not be confusing. The book repeats on p. 162 the legend that Euler named the constant \(e\) after himself.

Bottom line: Wait for the second edition, in the hope that the rough edges will be smoothed out. A good alternative with roughly the same prerequisites and coverage but different enrichment activities is Ross’s Elementary Analysis: The Theory of Calculus.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.