My main mathematical interests are in number theory and the history of mathematics. So what was I doing teaching Real Analysis? We do that sometimes, my colleague Ben Mathes and me: I teach Analysis, and he gets to teach Algebra. We have fun and vary our course assignments a little bit, and the students get the subliminal message that mathematics is still enough of a unified whole that people can teach courses in areas other than their own.
I had taught Colby’s Real Analysis course once before. The first time I teach a new course, I tend to just dive in and see what happens (it went all right). The second time, however, is when one starts to want to think about the course. This article is one of the results of that process. It may be that everything I have to say is well known to anyone who specializes in analysis; if so, I’m sure they’ll write in to tell me that. Still, maybe I can share a few insights.
Analysis courses can vary a lot, so let me first lay out the bare facts about our version. Real Analysis at Colby is taken mostly by juniors and seniors, with a sprinkling of brave sophomores. It is a required course for our mathematics major, and it has the reputation of being difficult. (This course and Abstract Algebra contend for the “most difficult” spot.) The content might best be summarized as “foundations of analysis”: epsilonics, the topology of point sets, the basic theory of convergence, etc. As the title of a textbook has it, the goal of the course is to cover “the theory of calculus.”
The first thing to do was to choose a textbook. The most common choice at Colby has been Walter Rudin’s classic, Principles of Mathematical Analysis. It is a hard book for students to read, but reading such books is a good skill for a mathematics major to acquire, and Rudin’s book repays the effort that students need to put into reading it. There is a lot of beautiful mathematics in it, and students eventually come to respect, perhaps even enjoy, the book.
But two serious problems have developed, neither of which is really Prof. Rudin’s fault. The first is the price. Principles is now so expensive that I feel guilty making students buy a copy. The second is the internet. Many instructors around the world have used this book, and in the goodness of their hearts have posted solutions to some of the problems for their students. They have neglected to set up their sites so that only their students have access to them, however. As a result, almost any problem from Rudin’s book has a solution online somewhere, and Google will find it for whoever wants it. Since I want my students to think about hard problems rather than learn to find their answers online, I decided I should look for another text.
To my rescue came the MAA Online book review column. Steve Kennedy had written a glowing review of Understanding Analysis, by Stephen D. Abbott (you can see it at http://www.maa.org/reviews/understand.html). The first paragraph of that review went like this:
This is a dangerous book Understanding Analysis is so well-written and the development of the theory so well-motivated that exposing students to it could well lead them to expect such excellence in all their textbooks. It might not be a good idea to create such expectations. You might not want to adopt this text unless you’re comfortable teaching from a book in which the exposition will nearly always be clearer than your lectures. Understanding Analysis is perfectly titled; if your students read it, that’s what’s going to happen.
And the price is very reasonable. So I decided this would be my textbook. I also decided to encourage students to buy a supplementary book, and made two suggestions: A Course of Modern Analysis, by Whittaker and Watson, and A Primer of Real Functions, by Boas. As I explained to them, one can’t imagine two more different books, and both of them are also very different from Abbott. Plus, one can buy all three for about the cost of buying Rudin’s Principles.
Now, I make no pretense of having carefully examined the available textbooks, which is why this isn’t a “what’s the best textbook” article. I went with what I knew, and with Steve Kennedy’s recommendation. It worked out pretty well, though next time I’ll probably want to choose different books for the supplement slot.
OK, so we launch into the course itself. Before we start, it is useful to ask what we want to achieve. The first answer comes easily: we want to introduce students to epsilon-delta proofs, and to use this technique to prove all those theorems they took for granted in their calculus course. But the answer generates questions of its own. Why do this? What do students gain from such knowledge?
That’s easy to answer if your students are likely to be going to graduate school. But most of my students are not, at least not immediately or not in mathematics. (A vast majority of Colby students do end up getting an advanced degree, but that’s different from going on to get a Ph.D. in mathematics.) So the question becomes a little sharper. I decided to take it for granted that my students wanted to learn mathematics, irrespective of their future career plans. The sharper question is, then, are epsilon-delta proofs a crucial thing for them to learn, and, if so, why?
Now, it’s easy to mount an argument that learning this stuff is in fact not that important. After all, lots of people learn and use sophisticated mathematics without ever having felt the need to delve into the foundations of the calculus. Convergence questions can usually be settled by “this gets small, and that’s even smaller” arguments. Turning those into formal epsilon-delta arguments is a nice party trick, and one that professional mathematicians have to know how to do, but no one should get too excited about it.
As a counter to that, let’s note that some of my students were planning to go to graduate school, and they would need to know the trick. And the course is there in the curriculum, and listed as required. So it must be important after all. This makes it clear that in order to justify the formalism of real analysis, one needs to find situations and problems in which the epsilon-delta approach is essential.
There is, of course, a very famous moment (by no means the only such moment, but probably the best known) in the history of mathematics that serves as an example: Cauchy’s “proof” that the sum of a series of continuous functions is continuous. As presented by Cauchy, this is a classic “this is small, and so is this” argument. And, of course, it doesn’t work, because hidden in the argument is a uniformity assumption. I decided to use this example (or a simplified version of it) fairly early on.
Abbott’s book turned out to be a good fit, because the author was clearly worried about similar issues. Each of his chapters begins with an example where things “go wrong” in ways that can only be understood by using the formal tools of analysis. Some of these are more convincing than others, but I was delighted to be using a textbook that noticed that such examples are crucial to the course.
One of the main threads in the course, then, was the idea of uniformity. The very definitions of uniform continuity and uniform convergence require the epsilon-delta formalism, and the notions simply cannot be understood without it. Over and over, I showed students examples of things that went wrong because something failed to be uniform, and showed them how to fix it by adding the uniformity assumption.
To this, I added two other threads, both stolen from articles written by wiser mathematicians than me. The first was the problem of partitioning the real numbers into two disjoint sets A and B, and then finding two functions f and g where f is continuous on A and discontinuous on B, and g does the reverse. The easiest case is when A is a singleton set (well, A empty is even easier, I guess), and students are usually able to push that a little. The case A = Q is the clincher: in this case the function g exists, but not the function f (i.e., no function can be continuous at all rational points and discontinuous at all the irrationals). This follows from a beautiful (and fairly easy) theorem of Volterra that William Dunham wrote up in an article for Mathematics Magazine some years ago. (The precise reference is given below.)
This thread doesn’t really connect to the issue of uniformity, but it fits in well with the point set topology. I also like it because it shows that not every kind of monster exists. What I mean is this: when students first start learning point set topology, they get a feeling that things can get arbitrarily pathological, that anything can be done. Cantor sets and similar objects tend to reinforce that feeling. Volterra’s theorem, however, shows that in fact not everything can be done, and it does it without having to introduce something like Baire category theory.
The second added thread is the problem of constructing a “nice” function that interpolates the factorials. In other words, I posed to students the question of how one should define x! when x is not an integer. David Fowler wrote a series of articles for the Mathematical Gazette on this topic, and I stole his ideas without remorse. For example, it’s easy to construct a continuous function that does the trick: define x! to be 1 if x is between 0 and 1, and then extend it by using the functional equation x! = x(x-1)!. Unfortunately, the resulting function fails to be differentiable. Finding the simplest way to define x! on [0,1] so that the extended function is differentiable is a nice exercise. Eventually, of course, we ended up with the Gamma function (on the reals). To do that, we had to deal with improper integrals depending on a parameter. (Unfortunately, Abbott doesn’t cover improper integrals and integrals depending on a parameter, so I had to fish this part out from other texts.) The nice thing is that for most properties of the Gamma function (continuity, differentiability, etc.) we had to deal with issues of uniformity again, neatly closing the circle. In fact, “as long as the convergence is uniform” came up so often in the last few classes that it became clear to me that this was the core concept I was teaching.
What did my students learn? By the end of the semester they certainly had a good enough understanding of uniform convergence and of the techniques used to understand and prove things related to it. I don’t know that that understanding will last very long unless it is reinforced in other courses. But they did understand, I think, that the difficult formalism of analysis is there for a reason, and that it is needed if we are to do interesting things with functions. I don’t think they’ll forget that. And that’s good enough for me.
“A Historical Gem from Vito Volterra,” by William Dunham. Mathematics Magazine, 63 (1990), pp. 234–237.
“A Simple Approach to the Factorial Function,” by David Fowler. Mathematical Gazette, 80 (1996), 378–380.
“A Simple Approach to the Factorial Function — the next step,” by David Fowler. Mathematical Gazette, 83 (1999), 53–57.