Tuesday, February 26, 2008


I am testing Driveway, a file sharing site. Our three FREE textbooks are there at the moment and I want to see if this is a reasonable way to share files. These are PDF files designed for on-screen viewing.

[BBT] is Real Analysis (Bruckner, Bruckner, Thomson) a graduate level real analysis text.
[TBB] is Elementary Real Analysis (Thomson, Bruckner, Bruckner) an undergraduate level real analysis textbook.
[TBB]-dripped is ... well something else.

If this doesn't work easily then ignore. I had one complaint about an ad on MediaFire that we have been using up to now. If anyone has recommendations on file sharing sites suitable for sharing mathematics textbooks let me know.

Thursday, February 21, 2008


#7. "The old man who refuses to travel in a train."

I should like to add a few words, in thanking Mr. Carey Francis for his paper. I hope the members of the Association realise that in order to be a serious mathematician it is necessary to have some knowledge of modern theories of integration. To be a serious pure mathematician and not to use the Lebesgue integral is to adopt the attitude of the old man in a country village who refuses to travel in a train.

It is necessary to learn these things and to get rid of the sort of terror which they appear to engender. It is true that the Lebesgue integral is very much easier than the Riemann, though naturally the beginnings of it are bound to be a little more difficult. And it is true, in a sense, that the Riemann integral is riddled with awkwardness and exceptions, but when one gets beyond the root of the subject, then the integral of Lebesgue is not really that of generalisation, but of simplification.

[These remarks were made by the President of the society after the presentation of the paper
MODERN THEORIES OF INTEGRATION. by E. C. FRANCIS, M.A., Fellow and Lecturer of Peterhouse, Cambridge in 1926. The paper itself along with these remarks can be found in the Mathematical Gazette, Vol. 13, No. 181 (Mar., 1926), pp. 72-77.]

Friday, February 8, 2008


What would Sir Isaac say?

Newton died in 1727 so contacting him about Riemann
(1826 – 1866) was not easy. But, with some help from a single malt scotch and a ouija board, I have managed to channel him. Here is the session, as best as I can recall.

He first gave me his definition of the integral.

Classical Definition. A function f has an integral on an interval [a,b] provided there is a function F defined on the interval and a subset N of the interval for which
  • N is a finite set
  • f is defined at all points of [a,b] except possibly points of N.
  • F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
  • F is continuous at points of N.
In that case the value of the integral is assigned to be F(b)-F(a).

Modern Definition
. A function f has an integral on an interval [a,b] provided there is a function F defined on the interval and a subset N of the interval for which
  • N is a set of measure zero.
  • f is defined at all points of [a,b] except possibly points of N.
  • F'(x)=f(x) at all points x of [a,b] expect possibly points of N.
  • F has zero variation on N.
In that case the value of the integral is assigned to be F(b)-F(a).

[A brief explanation. Only the notion zero variation on a set may be unfamiliar. You can find it in our dripped analysis text. N is a set of measure zero if the identity function F(x)=x has zero variation on N. The connection between the two definitions is that F has zero variation on a finite set N if and only if F is continuous at each point of N.]

-----------------------transcript of ouija board session---------------------

Me: Is this the correct definition of the integral?

Newton: Certainly. Integration on the real line is antidifferentiation. You can characterize that in numerous other ways, but that is what it is and that should be disclosed to students up front.

Me: Your definition has been called descriptive.

Newton: Indeed. It describes exactly what an integral is. Don't get confused by the fact that there are constructive versions. Start with this and get the constructions later.

Me: Why two versions?

Newton: For the 18th century the ``finite exceptions'' version was adequate for all applications. It was only with Fourier's introduction, in the 19th century, of series representations by trigonometric series that the ``measure zero exceptions" version became needed. Teach the former to your integral calculus students and save the subtle version for later.

Me: You say "get the constructions later." Who did that?

Newton: First there is Cauchy. He clarified the notion of continuity. We had a vague idea before, but he nailed it and showed that continuous functions have integrals. He also characterized the situation for functions with finitely many discontinuities.

Me: And ...

Newton: Certainly Lebesgue. He developed all the tools to characterize, construct the integral, and to investigate the class of absolutely integrable functions [i.e., both f and |f| are integrable].

Me: And ...

Newton: Well that leaves only Denjoy who gave an elaborate treatment for the constructibility of the integral of the nonabsolutely integrable functions.

Me: What about the Riemann integral?

Newton: A minor observation. Also misleading. It pushed integration theory in the wrong
direction for too long. His so-called "integral" only works for bounded functions and derivatives need not be bounded. Also it doesn't handle even bounded derivatives. As soon as that was realized then Lebesgue took it up as a problem: How to integrate constructively all bounded derivatives. He succeeded, of course.

Me: About the Henstock-Kurzweil integral, do you ....

Newton: That's just another descriptive integral. It looks kind of constructive, only because it is so similar to the Riemann integral. Its equivalent to my integral. In fact you can show it is included in my integral in a few short steps. Some people think you could start teaching integration theory with it, but I much prefer my own integral as a starting point. As I said, integration on the real line is antidifferentiation. Start there ...

------------end of transcript----------------------------------
[Sorry I don't recall any more, except that he began railing against Leibnitz, Locke, Pepys and some others.]

Wednesday, February 6, 2008

Having a hard time with proofs

Since making free PDF copies of our real analysis texts on our site ClassicalRealAnalysis.com we get an occasional appeal for help from users of the texts.

Here is the letter to us and my response.

R---- writes:

I happened across your book while doing some research on textbooks that could help me learn real analysis.

I've found that a lot of the topics I run into while working in AI and finance requires an understanding of real analysis. I've taken a significant amount of mathematics, statistics, differential equations, MVC, and linear algebra.

However, I'm finding myself getting stuck in ruts while trying to prove a lot of the exercises in the first chapter! Needless to say, it's quite frustrating when you can obviously see that it's true but can't formulate it into a proof. I realize that part of the battle to becoming good at math is being able to formulate the proofs on your own.

I was wondering if there were any hints or tips or books that I could read to help me with my inexperience in creating proofs.

Lastly, I greatly thank you for putting your book up online. I don't think I would have found it otherwise.

Thanks, R-------


My response:

Hi R------,
Don't be discouraged. I would say 95% of my students over the years would have said the same thing about proofs.

The Elementary Real Analysis text has an appendix where we talk about this. Certainly go there to be sure you know about Indirect Proofs, Direct Proofs, Contraposition and Proofs by Induction.

I would often for an exam announce in advance that a full proof would be required and tell the students explicitly which theorem. Even then many failed. So a first step is
to pick a theorem you like and memorize the proof and present that proof to as many people as are willing to listen. [I think of it like a jazz solo. If you want to learn how to improvise, first memorize someone's nice solo and perform it as often as you can. Your creativity won't follow until you learn first how to imitate.]

I am posting this on the alt.math.undergrad site and I'm hoping that others will have suggestions on how they managed to make this transition. Its way too many years back for me to remember how I managed, but I suspect it was because we studied Euclid in high school and had to produce hundreds of proofs in that style.