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Real Analysis: A First Course

Russell A. Gordon
Publication Date: 
Number of Pages: 
[Reviewed by
Marion Cohen
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“In a nutshell, this book presents the topics of a first-year calculus course, with all of the proofs and without the applications.” This is the one-sentence summary given by the author on p. viii, and it sounds like Heaven — as it sounded like, and was, Heaven several decades ago when I took what was then called “Advanced Calculus” back in undergraduate school. Reading this book brought me back to Heaven.

To me it has the perfect attitude. It’s written by somebody who obviously appreciates rigorous math, to the extent of sharing with his readers and students just what he appreciates. From the page 52 definition and explanation of what it means for a sequence to converge to a number — “The crucial point is that epsilon can be any positive number” and “It is worth pointing out that epsilon greater than zero comes first, then an appropriate N is sought” — to later more advanced concepts, such as the passage on page 245 which motivates uniform convergence: “It appears that pointwise convergence does not preserve any useful properties of functions. Since it is often necessary that the limit function inherit some properties, we will introduce a type of convergence that is stronger than pointwise convergence in the next section”. And in between, on page 212, after introducing geometric and telescoping series: “Since the convergence of an infinite series depends on the convergence of its sequence of partial sums and since sequences have already been studied in detail… it might seem as though there is little left to do. However, the two examples that have been given thus far have been misleading. For both examples, it was possible to find an expression for sn that did not involve a sum, then use this expresison to find the limit of the sequence. it most cases involving infinite series, it is not possible to express sn in a form which makes the limit easy to find.” These are the kinds of things that students who are not yet very familiar with “pure math” need to read and hear, and often don’t.

Also, I’m partial to authors and teachers who write or say, as Gordon does on page 257, “…a power series is an infinite degree polynomial” — to me, that’s the beauty of power series. I’m even more partial to authors who continue, “such series possess some nice properties that the typical series of functions does not possess.” This helps students to understand some of the subtleties — e.g., that indeed power series are series of functions, not “merely” numbers.

My very-favorites are the “Further Topics” and “Miscellaneous Results” sections, three of them appearing at the end of the chapters on “Differentiation”, “Sequences and Series of Functions”, and “Point-Set Topology”. E.g., it’s always exciting to find out, once again, that an everywhere continuous functions does not need to have a derivative, anywhere.

Since the book is for math majors, it probably doesn’t vitally need to be writen all that clearly. But it is. The motivations, explanations, definitions, and proofs are all extremely beautifully written. While being friendly and kind, he is definitely writing to math majors, and he gives them due respect. For example, he shares with his readers his perceptions concerning some of the proofs. Page 279: “This proof of the Weierstrass Approximation Theorem does not involve any deep ideas, but it is not all that enlightening. The origin of the polynomials is not clear and the convergence of the polynomials to the function is difficult to visualize…. Nevertheless, a sequence of polynomials that converges uniformly to a continuous function f on an interval [a, b] does exist.” (Actually, I disagree with the first statement. I did my Master’s and part of my doctoral dissertations on Schwartz distribution theory, and if I remember correctly, parts of the proof which he refers to seem very familiar; I think they relate to something called “delta-convergent sequences” and involve the notion of convolution products and how convergence sometimes preserves them. At any rate, armed with this background, this proof does seem enlightening, and it is indeed possible to, in some sense, “visualize” the sequence of polynomials and why they do what we want them to do. — What does “visualize” mean , anyway, in the context of math?)

Another good thing: He continually relates this “advanced calculus” to “non-advanced calculus”. E.g., page 142: “… the next theorem illustrates one such application. It is known as a monotonicity theorem since it gives conditions that guarantee a function is monotone. Its conclusion is probably familiar to you.” In fact, this book makes me wonder why “regular” calculus books are so often not clearly written; I firmly believe that math can be clearly written up, even without giving rigorous concepts and proofs, and even for students who are not “math people”. When I teach calculus, I make mention of clarifying ideas, and I give sketches of proofs, or sometimes simple statements that make things believable to “laypersons”. Probably other teachers do that also, and from my interaction with them, I’d venture a guess that they’d feel that their jobs would be easier, on both them and the students, if calculus books were better written.

Also, the author keeps abreast of the polarity between pure and applied, and he does it in just the same way that I would! E.g., page 130: “Although the deriva- tive concept has a number of practical applications (some of which should be familiar to the reader), the focus in this book will be on the mathematicial aspects of the derivative. However, a few simple applications of the derivative will appear in the exercises. (My own personal feeling is that, if teaching from this book, I would not specifically assign those exercises; I would just encourage the students to look at them — or perhaps during class, I might spend two or three minutes pointing them out.)

He also keeps abreast of the polarity between intuition and rigor. E.g., page 135: concerning the fact that the inverse of a continuous function is continuous: “…this result is clear from a geometrical perspective since the graph of the inverse function … is just the reflection of the graph of the function… through the line y = x. … as usual, this geometric reasoning does not constitute a proof that inverse functions are differentiable; the proof must use the definition of the derivative.”

Finally, he communicates perspective. Page 144: “Although l’Hôpital’s Rule is sometimes useful, it is not very important as a theoretical rool in real analysis. For this reason, it will be covered lightly here…”

He also communicates subtlety. In particular, his epsilon-delta stuff is a charm! Page 112: ”…it makes no real difference in this case if the inequalities are strict or not.”

I would now like to share a pet peeve. As a writer of poetry and creative non-fiction, I probably underline too much! But if I were a writer of math books, I would probably underline even more too much (as proven by the italicizing in this sentence…) In writing out this review, in fact, I was very very often tempted to underline certain words. It truly does seem to increase the clarity. My pet peeve is that editors and authors shy away from underlining, as though it were some sort of sin, or perhaps “cheap” in some way. What, I ask, is wrong with underlining? Especially in a math book. It would, again, make things clearer, and also serve to further emphasize the beauty of it all.

The author explains things in very much the same way that I would, and it was a challenge to try to think of ways in which I would change the book if it were mine. ( I always say that when I review a book which I love…) Here are two attempts:

Page 114: “Let f be a continuous function defined on an open interval (a, b). Then f is uniformly continuous on (a, b) if and only if f has one-sided limits at a and b.” I’d state that differently. In accordance with the preceding motivation, I’d say “f has one-sided limits at a and b (and therefore can be extended to a continuous function on all of [a, b]) if and only if f is uniformly continuous on (a, b).” That is, I’d make three changes: I’d change the order of the clauses, I’d throw in the parentheses, and I’d underline “uniformly ”. This, to me, would better convey the state of affairs.

Page 296: “There is no simple characterization of a closed set in terms of closed intervals”. This made me pause briefly, and might make some students pause less briefly. One could, of course, take the “dual” of the preceding theorem — “Every nonempty open set of real numbers can be expressed as a countable union of disjoint open intervals.” — and that would express any given closed set as a countable intersection of closed intervals. It think that what he means to say is: “There is no simple characterization of a closed set as any kind of disjoint union of closed intervals.”

I also have my own pet ways of teaching the material on pages 143-4. I call what the Second Derivative Test accomplishes “figuring out which and whether”. (whether a critical point is actually an extreme point and if so, which type it is, max or min). And I introduce L’Hôpital’s Rule by saying, “We’ve already seen derivatives via limits; now we’re going to see limits via derivatives.” And I call that rule “diff-ing across the board”. Since it’s unlikely that I’ll ever write a calc text, anybody is welcome to use those little gimmicks…

These are pretty nit-picky, I admit. And my two bigger complaints are also meant to be minor, in the light of so much major. First, I think that Chapter 5 on “Inte- gration”, in particular Sectons 5.1 and 5.2, could be more clearly written. That’s the stuff that tends to trip students up, or at least make them feel saturated. In fact, the author admits that it’s hard. So I think that this is the place for the author to show off, even more, his skills at motivating and making things visual. For example, page 166: The Norm of a partition is the length of the largest sub- interval; he should say that. Also, the norm of a tagged partition does not make use of the “tags”. Finally, on that same page, when he gives the example of a partition “tagged” with the midpoints of each subinterval, he might mention that this corresponds to the familiar Midpoint Rule. However, I commend his statement on the next page (167), where Riemann integrability is definited: “Once these subintervals have been chosen, the tag from each subinterval may also be chosen at random. As a result of all this variability, it is tedious and/or difficult to prove that a function is Riemann integrable on an interval using the definition unless the function has a very simple form”. But I also think that some of the more complicated proofs might be better motivated and illustrated (even as I agree that, at some point, math students should and will be able to grasp ideas with less motivation and illustration).

The only other “bigger” complaint — and this might be more of an opinion that a complaint — concerns Chapter 8 on “Point-Set Topology”. To me, topology isn’t topology without the core dea of open and closed sets (or some other equivalent core idea) — that any set, along with a collection of subsets which are closed with respect to finite intersection and arbitrary union, is a topological space. I read every single word in this chapter (and in this book) and I didn’t see anything to that effect. (Perhaps I missed it. He does, sometimes, refer to the “advantage” of proofs which use only the notions of open and closed sets rather than the properties of real numbers, but does not, in my perception, quite make it clear why this is so.)

I agree that the emphasis in this book is on real numbers, that the topology chapter should be treated accordingly, and that very little attention needs to be paid at this point to the abstract notions in topology. However, I believe that the author missed an opportunity. On page 294, he gives us “Theorem 8.4:

  1. The unon of any collection of open sets is open and the intersection of any finite collection of open sets is open.
  2. The intersection of any collection of closed sets is closed and the union of any collection of closed sets is closed.”

This theorem refers to open and closed sets of reals, but the author might, after proving this theorem, briefly say something to the effect that (“for the record”, as he often aptly says) this is the essence of rigorous point- set topology — that whenever a collection of subsets has property (a), they can be called “open” and the “universal” set a “topological space”, whether it consists of real numbers or not. After all, students reading this book either will be taking or already have taken Abstract Algebra, so they’re not totally unfamiliar with abstraction. Again, this might be more of a personal preference that anything even remotely serious.

All that said, this book most definitely gives us all an extremely illuminating picture of the spirit of real analysis, and of math. Students will work hard in the course and will be in Heaven.

Besides being a mathematician, Marion Cohen is a poet and writer, author of several books, the latest of which is Dirty Details: The Days and Nights of a Well Spouse (Temple University Press, PA). Her forthcoming book, Crossing the Equal Sign (Plain View Press, TX), consists of poetry about the experience of mathematics. You might have seen some of these poems in The American Mathematical Monthly. Marion can be emailed at and more of her writings can be seen on her website at .

1. Real Numbers.

What Is a Real Number?

Absolute Value, Intervals, and Inequalities.

The Completeness Axiom.

Countable and Uncountable Sets.

Real-Valued Functions.

2. Sequences.

Convergent Sequences.

Monotone Sequences and Cauchy Sequences.


Supplementary Exercises.

3. Limits and Continuity.

The Limit of a Function.

Continuous Functions.

Intermediate and Extreme Values.

Uniform Continuity.

Monotone Functions.

Supplementary Exercises.

4. Differentiation.

The Derivative of a Function.

The Mean Value Theorem.

Further Topics on Differentiation.

Supplementary Exercises.

5. Integration.

The Riemann Integral.

Conditions for Riemann Integrability.

The Fundamental Theorem of Calculus.

Further Properties of the Integral.

Numerical Integration.

Supplementary Exercises.

6. Infinite Series.

Convergence of Infinite Series.

The Comparison Tests.

Absolute Convergence.

Rearrangements and Products.

Supplementary Exercises.

7. Sequences and Series of Functions.

Pointwise Convergence.

Uniform Convergence.

Uniform Convergence and Inherited Properties.

Power Series.

Taylor's Formula.

Several Miscellaneous Results.

Supplementary Exercises.

8. Point-Set Topology.

Open and Closed Sets.

Compact Sets.

Continuous Functions.

Miscellaneous Results.

Metric Spaces.

Appendix A. Mathematical Logic.

Mathematical Theories.

Statements and Connectives.

Open Statements and Quantifiers.

Conditional Statements and Quantifiers.

Negation of Quantified Statements.

Sample Proofs.

Some Words of Advice.

Appendix B. Sets and Functions.



Appendix C. Mathematical Induction.

Three Equivalent Statements.

The Principle of Mathematical Induction.

The Principle of Strong Induction.

The Well-Ordering Property.

Some Comments on Induction Arguments.