This is truly a student-oriented text on single-variable Advanced Calculus, based on refinements of correspondence texts that have been used by more than 10,000 students in Britain’s Open University since 1971.

The first chapter (‘Numbers’) provides basic background on the real line, including a discussion of inequalities, without introducing any formal topology (compactness, etc.) The book uses a sequential approach to continuity, differentiability, and integration. Null sequences and their properties are introduced and are then used to discuss convergence of general sequences: A sequence {a(n)} converges to *L* if and only if {a(n) – *L*} is a null sequence. Continuity of a function *f* is defined in terms of the behavior of *f* on convergent sequences. After this kind of gentle preparation involving manipulation of only one epsilon and a (generally large) number *X*— not necessarily an integer — the student is introduced to the equivalent epsilon-delta definitions of limits and continuity. The epsilon-X and epsilon-delta methods are described as games (a strategy I have used): Player A chooses an epsilon and Player B tries to find an appropriate *X* or delta…. Proving convergence means that Player B wins. The *non*-convergence of sequences and functions is also considered.

Four appendices provide useful prerequisite information and solutions to problems (as opposed to ‘Exercises’, whose answers are not given). In between, most of the standard material of Advanced Calculus is treated in a way that even the ‘friendliest’ of current texts can only aspire to emulate. There are many helpful diagrams and marginal notes. Although I have often found such marginalia annoying in other texts, I see that in Brannan’s book these teaching suggestions, diagrams, and bits of advice to the reader (“You might like to compare this solution with that of Example 3 in Subsection 5.3.2,” “You may omit this proof at a first reading,”…) do provide a level of support and comfort to a student who may be reading the text on his or her own or in parallel with a formal course.

Among the gems scattered throughout, we find nicely motivated discussions of *π* and *e*, a detailed treatment of the ‘Blancmange’ function (Takagi fractal curve) — continuous everywhere but differentiable nowhere — a proof of Stirling’s formula, some theorems on the location of zeros of polynomials, and a proof that *π* is irrational. (Although the Index promises a discussion of the irrationality of *e* on p. 175, there doesn’t seem to be a proof anywhere in the book.)

Neither as elegant as Elementary Analysis: The Theory of Calculus by Ross or the recent books by Morgan, nor possessing the *gravitas* of the classic Advanced Calculus books (Apostol, Buck, Kaplan, …), Brannan’s book speaks to the average math or science student. The spirit in which this book has been written explains why the Open University was rated the top university in England and Wales for student satisfaction in the last two government national student satisfaction surveys. This text deserves adoption consideration by any instructor who wants to give his or her students a peek behind the curtains of real analysis to see some of its beauty and usefulness revealed.

Henry Ricardo ([email protected]) is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.

## Comments

## Henry Ricardo

This is an addendum to my published review. Brannan does indeed prove that e is irrational, but on pp. 125-126 (Theorem 2) of the paperbound edition of his text rather than on p. 175. Also, I should have mentioned that the author does not discuss Cauchy sequences at all.