Classical Real Analysis

New ClassicalRealAnalysis.info web site

2012-02-10T16:17:00.000-08:00

IMPORTANT NOTICE FEB 10, 2012

Our .COM web site is being redesigned and moving to a new server.

Go to

for all resources now.

The .com site will reopen as a commercial site for the sale of paperback copies of our texts.

Why the move: MS Office Live has hosted this service since 2008. Originally they offered free domain name and free hosting for small businesses. Last year they cancelled free domain name. This year they cancelled free hosting. Since this web site is not portable the way MS designed it, we have reworked everything and found a new host (Gigapros).

The new site:

ClassicalRealAnalysis.info

is still under construction but has all the FREE files available as well as instructions for purchase of paperbacks.

Come back to

ClassicalRealAnalysis.com

after April 2012 to see a leaner more commercial site. The .info site will be the site most students are looking for.

FREE PDF POLICY: Our new policy is to distribute completely FREE PDF files for all of our texts.

You can find the files on

ClassicalRealAnalysis.info

FREE PDF FILES (Mathematics)

2011-11-29T12:05:00.000-08:00

The YOUPUBLISH web site that we previously used to distribute our files appears to have dropped into a black hole. If you need any of our files please write to me thomson@sfu.ca and I will send you a link to access the files without charge. [November 2011]

The texts currently available are:

ELEMENTARY REAL ANALYSIS (undergraduate)
REAL ANALYSIS (graduate)
MATHEMATICAL DISCOVERY (freshman or senior high school)
THE CALCULUS INTEGRAL (post calculus [experimental])
THEORY OF THE INTEGRAL (specialist)

THEORY OF THE INTEGRAL

2011-11-14T16:31:00.000-08:00

New book to be launched in April 2012.

Theory of the Integral, Brian S. Thomson, ClassicalRealAnalysis.com (2012), [ISBN 1467924393]

Prepublication PDF file: FREE PDF FILE date Nov 27 2011

Watch for updates on our web page and please send comments to the author.

This text is intended as a treatise for a rigorous course introducing the elements of integration theory on the real line. All of the important features of the Riemann integral, the Lebesgue integral, and the Henstock-Kurzweil integral are covered. The text can be considered a sequel to the four chapters of the more elementary text THE CALCULUS INTEGRAL which can be downloaded from our web site. For advanced readers, however, the text is self-contained.

Book launch of "Mathematical Discovery"

2011-07-12T12:25:00.000-07:00

Mathematical Discovery
List Price: $15.49

Mathematical Discovery
Authored by Andrew M Bruckner, Brian S Thomson, Judith B Bruckner

This book is an outgrowth of classes given at the University of California, Santa Barbara, mainly for students who had little mathematical background. Many of the students indicated they never understood what mathematics was all about (beyond what they learned in algebra and geometry). Was there any more mathematics to be discovered or created? How could one actually discover or create new mathematics? In order to give these students some sort of answers to such questions, we designed a course in which the students could actually participate in the discovery of mathematics.

ISBN/EAN13: 1453892923 / 9781453892923
Page Count: 268
Binding Type: US Trade Paper
Trim Size: 7" x 10"
Language: English
Color: Black and White

Go to our web site
http://classicalrealanalysis.com/DISCOVERY.aspx
for FREE PDF FILES! Or purchase an inexpensive paperback from
https://www.createspace.com/3493013
Make sure to use the discount code MD2S9X2Q to get 20% off.

To Download BBT as an E-Book.

2011-04-07T14:19:00.000-07:00

To obtain a copy of our graduate level real analysis textbook,
REAL ANALYSIS, Bruckner, Bruckner & Thomson

NO LONGER USING WebDeliverySolutions.com ...Just go to our web site for download instructions.

To Download TBB as an E-Book.

2011-04-07T13:54:00.000-07:00

Trying out a new way of getting our PDF files to the end user.

So this is just an experiment. To purchase a E-book version (really just a huge PDF
file) of our undergraduate real analysis textbook:

ELEMENTARY REAL ANALYSIS by Bruckner, Thomson, & Bruckner.

NO LONGER USING WebDeliverySolutions.com. JUST GO TO OUR WEB SITE.

NEW BOOK--MATHEMATICAL DISCOVERY

2010-11-20T17:54:00.000-08:00

We are preparing a new book on Mathematical Discovery. It is intended for students with just the usual high school background in mathematics (algebra and geometry) who wish to learn how mathematicians discover new methods to attack problems. It is based on a course Andy Bruckner gave to students at UC Santa Barbara for non math majors, covering interesting topics that require less background preparation than the usual university-level courses would need.

There will be (as usual) FREE PDF FILES as well as a trade paperback version. The first chapter on Tilings discusses the famous problem of "squaring the rectangle," i.e., tiling a rectangle with a number of unequal sized squares. You can download this one now.

Riemann integral indeed?

2010-02-26T16:17:00.000-08:00

For fans of this BLOG and the DRIP crusade (yea, both of you) it may appear that I have gone to the dark side since my two most recent papers are studies of properties of the Riemann integral:

B S Thomson, "Monotone convergence theorem for the Riemann integral", American Math Monthly, Vol. 117, no. 6, June-July 2010.

B S Thomson, "Characterizing the indefinite Riemann integral", Real Analysis Exchange, 2010 [to appear].

You can download preprints from here and here respectively. I hope you will (both) grant me absolution if I promise to return to a polemic on this unfortunate integral. Please write.

FREE CALCULUS BOOK

2009-07-20T13:38:00.000-07:00

The Calculus Integral [Beta0.2],
B S Thomson,
ClassicalRealAnalysis.com (2009).

Download a free PDF file from http://www.youpublish.com/files/23072.

Well "calculus book" doesn't quite describe it. It is an account of integration theory on the real line that starts in the initial chapters with the "calculus integral" (i.e., the original integral of Newton) and carries the development as far as the integrals of Lebesgue and Henstock-Kurzweil.

If you are, however, teaching a calculus course and have ever considered dropping the Riemann integral and its ugly step-sister the improper Riemann integral from the syllabus, then you might want to use the first three chapters as a basis for the teaching of the basic integral of the calculus.

A trade paperback version will be available shortly.

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #5

2009-04-10T08:59:00.000-07:00

It clears up the mystery of the popcorn function.

Suppose a function is zero at every irrational point. What is its integral? Well, certainly, the points left out are insignificant. Not merely a set of measure zero but even countable. Such a set plays no role in determining the integral so your function can have any values on the rationals, or even remain undefined on the rationals. The simple answer is that your function is integrable on every interval with a zero integral.

Oh, what? Sorry? You have only learned the Riemann integral. Alas, the answer now is entirely different. Now the function must be defined at these missing rational points and the answer depends on how you define them. If the resulting function is integrable then certainly the value of the integral is zero, but it may or may not be integrable. Let xn be a listing of all the rationals and let your function be defined to be f(xn)=cn and with f(x)=0 at x irrational. What is a necessary and sufficient condition for f to be integrable [i.e., integrable in the dumb Riemann sense]? That's a tough question, but one that is not particularly important.

In 1875, K. J. Thomae discovered the now-famous example of a function of this kind that is continuous at all the irrationals and discontinuous at the rationals. This function has many names: the modified Dirichlet function, Thomae function, Riemann function, raindrop function, ruler function, and popcorn function.

His example is a nice curiosity in the study of continuous functions. But it is usually presented to students of integration theory as example of a seriously discontinuous function that is integrable. The student gets the impression that it is important to have continuity, that discontinuities must be controlled, that without proper configuration of the values of a function the integral is badly affected, and that integration theory has its mysteries. That's good teaching?

If we drop the Riemann integral then the popcorn function would not be mentioned in the context of integration theory and can return to its proper place in the study of continuity.

Is there a function continuous at every rational and discontinuous at every irrational?
Is there a function discontinuous at every rational and continuous at every irrational?

The answer to the first question is "no" and the answer to the second question is "popcorn."

Free books aren't free?

2009-02-20T11:10:00.000-08:00

One of our correspondents expressed the view that our free books didn't seem,
to him at any rate, to be free. Presumably because YouPublish asks for $1.00 to download PDF files.

But all of our books are fully viewable and searchable on Google Book Search, which you can try out below.

In addition most of the chapters of TBB (for example) can be downloaded without charge from our web site page.
The other issue, I suppose, is whether the $1.00 charge on YouPublish is irritating. Irritating, I mean, in the sense that your free Lone Ranger Decoder Ring (only $1.00 shipping and handling) was irritating when you were 12 years old.

Free digital texts begin to challenge costly college textbooks

2008-08-18T13:58:00.000-07:00

Would-be reformers are trying to beat the high cost -- and, they say, the dumbing down -- of college materials by writing or promoting open-source, no-cost online texts.

By Gale Holland, Los Angeles Times Staff Writer
August 18, 2008 gale.holland@latimes.com

The annual college textbook rush starts this month, a time of reckoning for many students who will struggle to cover eye-popping costs of $128, $156, even $198 a volume.

Caltech economics professor R. Preston McAfee finds it annoying that students and faculty haven't looked harder for alternatives to the exorbitant prices. McAfee wrote a well-regarded open-source economics textbook and gave it away -- online. But although the text, released in 2007, has been adopted at several prestigious colleges, including Harvard and Claremont-McKenna, it has yet to make a dent in the wider textbook market.

Bookstore

"I was disappointed in the uptake," McAfee said recently at an outdoor campus cafe. "But I couldn't continue assigning idiotic books that are starting to break $200."
...

FOR THE REST OF THE ARTICLE GO HERE>>>

Trade Paperback of BBT now available!

2008-07-31T14:19:00.000-07:00

[BBT] Real Analysis,
Second Edition (2008)
Andrew M. Bruckner, Judith B. Bruckner and Brian S. Thomson
ISBN:1434844129
EAN-13:9781434844125

This is the second edition of our graduate level real analysis textbook formerly published by Prentice Hall (Pearson) in 1997.

In addition to "nearly" free PDF files of this text that have been available for a while now, we are now offering a trade paperback version. This will soon be listed on Amazon, and is already listed on CreateSpace. For details on ordering or for download instructions visit our website.

YouPublish chooses us a "Most Intriguing Creation"

2008-07-08T15:40:00.000-07:00

from their July 8 emailing to their clients ...

Editor's Picks
Featured Creators

Most Intriguing Creation

[BBT] Real Analysis , 2nd Edition (2008)
By ClassicalRealAnalysis.com
With A.M. Bruckner, J.B. Bruckner, and B.S. Thomson

A YouPublish top-seller publishing downloadable versions of mathematics textbooks and driving traffic to their website.

Classical Real Analysis provides some relief to cash-strapped students and their families by offering digital textbooks, online free of charge or almost, or in hard copies for a small fraction of the cost of traditional books. The digital books can also be easily customized and updated, eliminating the waste experienced with traditionally published textbooks.

Publishers like Classical Real Analysis know colleges and universities must embrace new methods of textbook development and distribution if they want to rein in runaway costs. YouPublish is proud to support this movement toward affordability and sustainability.

Check out their works at:

youpublish.com/classicalrealanalysis

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #6

2008-05-17T11:54:00.000-07:00

#6. It's all that improper stuff.

im·prop·er (from Merriam-Webster Online Dictionary)
Pronunciation:
\(ˌ)im-ˈprä-pər\
Function:
adjective
Etymology:
Middle English, from Middle French impropre, from Latin improprius, from in- + proprius proper

: not proper: as (a:) not in accord with fact, truth, or right procedure : incorrect <improper inference> (b:) not regularly or normally formed or not properly so called (c:) not suited to the circumstances, design, or end <improper medicine> (d:) not in accord with propriety, modesty, good manners, or good taste <improper language>
synonyms see indecorous .

How is an integral improper? Well, if your main point of reference is the unfortunate Riemann integral then, in any situation in which you need to perform an integration of a non-Riemann integrable function, that can only be considered "improper". Thus calculus students are drilled on the need to interpret all "proper" integrals as Riemann integrals. If a function is unbounded then it cannot have an integral but it might have an "improper" integral.

In all advanced mathematics the class of integrable functions includes an abundance of unbounded functions. It is a feature of the theory. There is nothing improper about such integrals because the definition of the integral includes them. All of the calculus drill on handling unbounded functions is dropped for advanced classes. But, in most cases, we still teach beginning students using the terminology and the methods of the nineteenth century.

The Riemann integral should be dumped if only to get rid of all this improper nonsense.

It even has its particularly stupid aspects. Suppose we wish to integrate f(x)=x^(-1/2) on the interval [0,1]. The fundamental theorem of the calculus says search for a suitable antiderivative, and F(x) = 2 x^(1/2) comes to mind fairly soon. So the integral has the value F(1)-F(0)= 2.

"But", howls the calculus professor, "You didn't notice that the function f(x) here is unbounded, therefore it has no integral properly speaking. You should have integrated on [t,1] for all t between 0 and 1 and then taken the limit as t tends to zero on the right. Then you must assert that the function does not have a proper integral but does have an improper integral equal to ...well yes, the same value 2 as before, but this is the correct procedure that must be followed."

That Book Costs How Much?

2008-04-25T17:36:00.000-07:00

New York Times Editorial

That Book Costs How Much?

Published: April 25, 2008

Colleges and universities will need to embrace new methods of textbook development and distribution if they want to rein in runaway costs.

...for the full article use this link.

Amazon offers [TBB] at a 28% discount.

2008-04-19T10:36:00.000-07:00

Elementary Real Analysis: Second Edition (2008) by
Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner
(Paperback - April 7, 2008)

Buy new: List price $33.95. Amazon discount price: $24.40

Get it by Tuesday, April 22 if you order in the next 49 hours and choose one-day shipping.

Eligible for FREE Super Saver Shipping.

Anyone know how Amazon decides on such things? Is there some algorithm they use that every now and then pops out a discount on one of their listings? Are they partial to mathematics students and just want to offer cheaper books for them? [Not likely, my Marcel-Dekker book is listed at about the same price as a Mouton Rothschild.]

[TBB] - fat and not so fat!

2008-04-14T19:55:00.000-07:00

The full version of [TBB] Elementary Real Analysis, Thomson-Bruckner^2
pp. 684, is available (as the last post indicates) on both E-Store and Amazon.

Some users may prefer the smaller versions.

Volume One is a nice handy size at 408 pages and can be ordered too [HERE]. It includes Chapters 1-8 and is suitable for about two semesters of undergraduate real analysis. Volume Two is available [HERE] and includes Chapters 9-13, with 290 pages.

As always you may simply download a FREE PDF file containing all of the [TBB] material in an form optimized for on-screen viewing by visiting our web site.

[TBB] paperback version now available.

2008-04-11T09:19:00.001-07:00

The second edition of our text, [TBB] Elementary Real Analysis (Thomson/Bruckner/Bruckner)
is now available as a trade paperback. For ordering information through E-store or Amazon.com please go to our web site: http://classicalrealanalysis.com/download.aspx

This is actually a pretty large paperback (7" x 10" x 1.7"). If you are a student having to carry it around between classes you might prefer to wait for the split version (Volumes I and II) that will be up shortly.

If you are on a budget... Well remember that all of our titles are there as FREE PDF files in a version optimized for on-screen viewing.

[BBT] Real Analysis, Second Edition (that's the graduate level textbook) can be downloaded now too, and will be available as a trade paperback in Summer 2008.

FILE SHARING --- JUST TESTING

2008-02-26T17:05:00.001-08:00

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.

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #7

2008-02-21T15:06:00.000-08:00

#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.]

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #8

2008-02-08T09:18:00.000-08:00

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.]

Having a hard time with proofs

2008-02-06T10:32:00.000-08:00

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.

********************************

Designing On-Screen Versions of Mathematics Texts?

2008-01-31T16:48:00.000-08:00

Help 1?
I have redesigned the on-screen versions of our real analysis textbooks. They are here
if you want to download them. BBT is the graduate level real analysis, TBB is the undergraduate level real analysis and TBB-Dripped is ...well something else.

Open one of these in Adobe Reader, resize the reader, resize the fonts, and open the bookmarks. Is this design workable on your screen? Pick your favorite real analysis topic, look it up in the index, click to see the text and scan a couple of pages. Any suggestions?

Help 2?
Specifically, does anyone know how to work around the fact the the LaTeX packages endnote and hyperref ignore each other? That is one of the bugs in my on-screen version. SEE NOTE in yellow doesn't link forward to the hint in the NOTES. I do have a link back though.

Help 3?
We are working on a PRINT-ON-DEMAND service for all three of these titles. Does anyone have stories, good or bad, to relate about CreateSpace, the new Amazon.com service?

[Some students in a class in the midwest used the on-screen versions for their assigned text, but most ended up finding used copies anyway. So, if this is a common reaction, we are trying to supply inexpensive trade paperback copies. Or, perhaps, we just needed a better on-screen version.]

The texts here are Elementary Real Analysis and Real Analysis by Bruckner, Bruckner and Thomson, previously published by Prentice Hall (Pearson) in 2001 and 1997. Full versions of these texts (recently corrected) are available as free downloads.

COVER DESIGN (cont. again)?

2008-01-29T10:21:00.001-08:00

This one is a little more conservative. Very elegant, but I think I prefer the other two.

COVER DESIGN (cont.)?

2008-01-29T10:16:00.001-08:00

I think this one is my favorite though. Anyone have an opinion that you would like to share?

COVER DESIGN?

2008-01-29T09:48:00.000-08:00

We are currently selecting our cover designs for the trade paperback copy of Elementary Real Analysis.

Anybody like this one?

[I am negotiating with my colleagues to correct the spelling of my name.]

OLD EXAMINATIONS AND ASSIGNMENTS

2008-01-28T11:35:00.000-08:00

I have posted some old examinations and homework assignments for courses in Elementary Real Analysis that I taught in the past. You can download them directly from here

[Download now]

or you can find them on the DOWNLOAD page of our website

http://classicalrealanalysis.com/download.aspx

There is a lot of material, with a high level of redundancy. If you are a student of real analysis at the undergraduate level it might be useful to try some of the examinations as part of your review. Any instructor who wishes to steal any of this material for their own use should likely ask for the LaTeX source file from me, since that will be a lot easier to pilfer from.

Top Ten Reasons for Dumping the Riemann Integral #9.

2008-01-19T18:02:00.000-08:00

#9. Does the phrase "mildly interesting exercise" suggest anything?

The quoted remark is from Jean Dieudonne, the French mathematician and well known Bourbakiste, in his dismissal of the Riemann integral as a suitable object of study for undergraduate mathematicians.

Actually I find the history itself more than mildly interesting and, with your indulgence, will give a fractured account of it here. What happened to Riemann follows a pattern that I can describe this way:

Suppose that you (Jones) discover an old theorem of Smith that you think you can improve on.

Theorem [Smith] Every object Y has the property Z.

You (Jones) decide that it might be worthwhile to characterize this property Z, especially since Smith seems to have gotten some fame from it. You succeed and write it up like this:

Definition [Jones] We say that any object possessing the property Z is a zamboni.

Theorem [Jones] A necessary and sufficient condition for an object to be
a zamboni is [insert your characterization].

Corollary [Smith] Every object Y is a zamboni.

Thus you (Jones) have, with a stroke of the pen, transformed Smith's theorem into a trivial corollary of Jone's theorem.

The success of Jone's manoever here is whether zambonis are going to have any lasting interest. If they do and prove to be a significant concept, then your fame exceeds Smith's, even though Smith had the original insight. If zambonis fall flat on the mathematical world then move on to something else.

Well, Riemann's zamboni is his integral. He took a theorem of Cauchy asserting that every continuous function on a compact interval [a,b] had a certain property with regard to its integral. He then characterized that property and gave it a name. Hence the definition of the Riemann integral, Riemann's characterization of integrability, and Cauchy's theorem dropping down to the status of a corollary. In fact the sums that Cauchy had used are now called "Riemann sums," so history has taken all the credit from Cauchy and shifted it to Riemann.

Well Riemann doesn't need that credit since he did far better things. Even the Riemann integral was just a throwaway in a 1854 paper about trigonometric series and not anything that he likely spent too much time thinking about. If he had seriously directed his enormous intellect at the problem of integration itself he would certainly have discovered the correct integral.

Unfortunately for Riemann, however, is the fact that this particular zamboni was misguided. Certainly it is a worthwhile professional project to characterize the property that Cauchy had discovered, but it was a sad mistake that future generations employed Riemann's integral as the central tool of integration theory.

For the next fifty years or so (1854--1901) mathematicians took Riemann's integral as if it were the correct one for bounded functions, and spent their time on the problem of integrating unbounded functions. Perhaps they thought of their program as "extending the integral to unbounded functions." Too bad. If you look at the problem this way you fail. If, instead, you tackle the problem of how best to integrate bounded functions, then the unbounded case takes care of itself. Enter the 20th century and Lebesgue. Lebesgue's thesis swept away all the previous theories.

At that point the history gets a bit strange. Lebesgue's theory is considered too difficult for some to learn and for many to teach. So many did not teach it, and many did not learn it. In the last 100 years there have developed many different ways to teach integration theory, some not any more difficult than teaching the Riemann integral itself. But, even so, we still don't teach it until graduate school at many places.

Google Base?

2008-01-18T10:07:00.000-08:00

I put one of the textbooks up on GOOGLE BASE.

http://base.google.com/base/a/briansthomson/3109186/D4317767210028435757

Does anyone actually use GoogleBase?

Top ten reasons for dumping the Riemann integal (#10).

2008-01-15T16:28:00.000-08:00

#10. Well...it's just so "uniformly" bad!

Indeed that's the problem. It is a uniform definition that many insist on teaching in an era when a pointwise definition is universally recognized as providing the correct theory of integration.

A function f(x) is integrable on [a,b] if for every e>0 there is a function d:[a,b] -> (0,1) so that when a partition is finer than d then the Riemann sum is ...

A function f(x) is Riemann integrable on [a,b] if for every e>0 there is a constant function d:[a,b] -> (0,1) so that when a partition is finer than d then the Riemann sum is ...

Why teach the uniform version? Well if you insist on it. Then perhaps you should also teach the uniform derivative too:

A function F on [a,b] has a uniform derivative
F'(x) = lim [F(x+h)-F(x)/h
if this limit exists uniformly for x in [a,b] as h->0.

Here is a completely trivial theorem. If F has a uniform derivative then F' is Riemann integrable. That's much easier than the usual version in calculus where you assume Riemann integrability and use the mean-value theorem.

Here's two more trivial theorems, with almost exactly the same proofs:

If f is continuous at each point of [a,b] then f is integrable.
If f is uniformly continuous on [a,b] then f is Riemann integrable.

But don't try proving that continuous functions are Riemann integrable. That's way harder. You have to first prove that continuous functions are uniformly continuous. Indeed here is the problem: to use the Riemann integral you are always obliged to check for a uniform condition. This imposes a burden in every proof, either a burden within the proof or a burdensome extra
hypothesis.

Teaching the Riemann integral quickly opens up this little puzzle of pointwise vs uniform conditions. Once this door is open you may as well come clean and own up to the existence of a nonuniform version of the integral that is universally used. Push that door shut to keep out any more of the theory than you are willing to teach, but don't suppress the whole story. You can tell them that the best way to teach this integral is not with this simple definition but by developing Lebesgue's theory of measure, and that only graduate students are considered mature enough to handle such sophisticated ideas. (It's a load of rubbish, but some people apparently believe it.)

It is well-written, user friendly, and free!

2008-01-15T08:58:00.000-08:00

In a discussion on the SciMath forum a poster named Alan comments

I am reading through Elementary Real Analysis by Thomson, Bruckner & Bruckner. It is well-written, user friendly, and free!

His question just arises from the use of the word "continuous" and the discussion flows around how to sort it out. This is a quite normal process in learning a subject as subtle (I wouldn't say difficult) as real analysis. The nomenclature has a strange power to occlude your reasoning at first.

But I like the endorsement. I hope Alan will encourage us to quote him in all our promotions. Although it is an odd activity, to promote an entirely free product, there is some great reward in knowing that your users appreciate both the effort and the price.

Trade Paperback copies of TBB

2008-01-14T11:29:00.000-08:00

We are currently planning on placing our undergraduate textbook, Elementary Real Analysis, on CreateSpace. In addition to the free PDF files that are currently available, this will mean that a Print-On-Demand service can be used to print inexpensive copies.

If anyone has advice or encouragement to offer in this project please send it along. At this stage we can imagine a multitude of things that can go wrong.

Any other titles available?

2008-01-12T16:00:00.000-08:00

Someone sent this message through the Contact Us form on our website:

>| Hey! is it at all possible that you guys are expanding
>| and can get other books available as well, or is this
>| the entirety of your collection. I am asking because
>| I'm looking for a book that you don't seem to have
>| at the moment Thanks -Schwinn

His return e-mail did not work. The answer (if you
are out there) is that what you see are all the titles
that we have available for distribution. We would be
happy to put up other titles in real analysis that could
be distributed for free.

Updated versions of our textbooks uploaded

2008-01-12T15:45:00.000-08:00

So far, since September, we have had over 9000 downloads
of our texts from the website:

             http://classicalrealanalysis.com/download.aspx

The files currently posted there have had a number of corrections made.  If you are using these for classroom instruction you might want to have the students access the most recent versions.

These are the undergraduate and graduate real analysis texts formerly published by Prentice Hall:

[TBB]  Elementary Real Analysis, Brian S. Thomson, Judith B. Bruckner,
Andrew M. Bruckner. Prentice-Hall, 2001, xv 735 pp. [ISBN
0-13-019075-61]

[BBT]  Real Analysis, Andrew M. Bruckner, Judith B. Bruckner, Brian S.
Thomson. Prentice-Hall, 1997, xiv 713 pp. [ISBN 0-13-458886-X]

In addition there is a new [experimental] version
of [TBB]  Elementary Real Analysis: Dripped Version.

This has an enlarged treatment of integration theory on the real line with the Riemann and improper Riemann integral removed, in alignment with the  D.R.I.P. program. Consider it a preliminary write-up.  Use at your own risk   :) 
[D.R.I.P. = Dump the Riemann Integral  Project]

Best wishes to all for 2008,
Brian Thomson

Classical Real Analysis

New ClassicalRealAnalysis.info web site

FREE PDF FILES (Mathematics)

THEORY OF THE INTEGRAL

Book launch of "Mathematical Discovery"

To Download BBT as an E-Book.

To Download TBB as an E-Book.

NEW BOOK--MATHEMATICAL DISCOVERY

Riemann integral indeed?

FREE CALCULUS BOOK

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #5

Free books aren't free?

Free digital texts begin to challenge costly college textbooks

Bookstore

Trade Paperback of BBT now available!

YouPublish chooses us a "Most Intriguing Creation"

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #6

That Book Costs How Much?

Amazon offers [TBB] at a 28% discount.

[TBB] - fat and not so fat!

[TBB] paperback version now available.

FILE SHARING --- JUST TESTING

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #7

TOP TEN REASONS FOR DUMPING THE RIEMANN INTEGRAL #8

Having a hard time with proofs

Designing On-Screen Versions of Mathematics Texts?

COVER DESIGN (cont. again)?

COVER DESIGN (cont.)?

COVER DESIGN?

OLD EXAMINATIONS AND ASSIGNMENTS

Top Ten Reasons for Dumping the Riemann Integral #9.

Google Base?

Top ten reasons for dumping the Riemann integal (#10).

It is well-written, user friendly, and free!

Trade Paperback copies of TBB

Any other titles available?

Updated versions of our textbooks uploaded