Thursday, December 27, 2012

The problem of characterizing derivatives

I noticed that someone had asked a question not too long ago on the MathOverFlow site about characterizing derivatives.  The answers were somewhat helpful but none of the usual experts seems to have dived into the discussion, so there was only limited information. 

I did add a comment, but maybe I should post some stuff here for those who happen to have some interest in the problem.

The problem naively is to find some necessary and sufficient condition in order that a function  f: [a,b] --> R  would be the derivative everywhere in [a,b] of some other function.   (For example continuity would be sufficient but certainly not necessary,  Baire class 1 would be necessary but not sufficient.)

To research the problem you should consult at least the following:

Andrew M. Bruckner, Differentiation of real functions. Lecture Notes in Math., vol. 659, Springer-Verlag, Berlin and New York, 1978, x + 246 pp.
or the more recent edition:
Differentiation of real functions.
Second edition. CRM Monograph Series 5.  American Mathematical Society, Providence, RI, 1994. xii+195 pp. ISBN: 0-8218-6990-6 

Chapter seven contains an entertaining and accessible account of the problem, including the original formulation of the problem by W. H. Young.

Andy has updated his discussion of this problem in a 1995 survey article written for the Real Analysis Exchange:   

Bruckner, Andrew M.  The problem of characterizing derivatives revisited.  Real Anal. Exchange  21  (1995/96),  no. 1, 112--133.
 Since then there have been a few varied expressions of conditions that characterize derivatives in perhaps unusual ways:
  1. On Characterizing Derivatives,  D. Preiss and M. Tartaglia Proceedings of the American Mathematical Society, Vol. 123, No. 8 (Aug., 1995), pp. 2417-2420
  2. Freiling, Chris,   On the problem of characterizing derivatives.
    Real Anal. Exchange 23 (1997/98), no. 2, 805–812.
  3. Brian S. Thomson, On Riemann Sums, Real Analysis Exchange Vol. 37(1), 2011/2012, pp. 1–22

Tuesday, December 18, 2012

Chinese translations of mathematics articles?

We are working on a Chinese translation of our textbook Elementary Real Analysis.   Chinese students can already download a free PDF in the English language version from our web site, but they will have to wait a bit for our translated version.

Oddly enough I just learned that the Chinese have begun a journal dedicated to translating articles from English to Chinese.  This is part of the letter where they inform me that they have translated a Monthly paper of mine.  (They received permission from the MAA -- I was not in the loop.)

"I am writing this letter on behalf of the editorial office of "Mathematical Advance in Translation", a translated journal which was sponsored by the Academy of Mathematics and Systems Science (Chinese Academy of Sciences). The journal is non-profit targeting at providing a general information about the world mathematics advances. Many of the appearing articles are translated papers that are well selected from "Notices of AMS" and "Bulletin of AMS", "The Amer. Math. Monthly" and "Mathematics Magazine" of the Mathematical Association of American(MAA), “SIAM News" and " SIAM REVIEW", “EMS Newsletter" and et al. From the very first issue, it always states very clear the originations, including author, title, journal, year, volume and pages, to ensure the reader easily referring to the original. The distribution is in a small scale, only around 600-800 subscriptions, all of them at a cost price, but the journal was very welcomed among the mathematicians, especially those in the rural areas and the young school teachers to them such kind of material are not easily accessible."

For the curious (and the bilingual) here is a link to my paper in English:

   Thomson, Brian S.
   Monotone convergence theorem for the Riemann integral. 
   Amer. Math. Monthly  117  (2010),  no. 6, 547--550.

and here is a link to a scan of the Chinese version that they published  (ISSN 1003-3092):

   [Chinese Language Version

I have no idea how accurate and literate the translation is.  Perhaps someone will comment?

Friday, November 9, 2012

A New Approach to College Textbooks. Finally.

That's not our slogan, but the slogan of

They did indeed come up with a useful (if not quite new) approach.  Their original idea (exactly like ours) was to offer free electronic versions of their textbooks as well as inexpensive paperback editions.

Our textbooks are just the ones we have written, and all of them are mathematics textbooks.  FlatWorldKnowledge hoped to have all college textbooks in all areas available in their model.

It's a great idea for the students.  Free PDF files.   If you also want a paperback, then buy one for a reasonable price (not the $100 or more the major publishers want).  It's also a great idea for authors.   The royalty from the big publishing houses is pretty small anyway, so why not bypass them and give the students a break.  Many students just use the free version -- some (enough) buy the paperback.

As it turns out, though, perhaps this "new" idea is not so commercially viable, or, if it is viable, it is just not profitable enough for most entrepreneurs. 

FlatWorldKnowledge has now announced: 

"Starting January 1, 2013, we will no longer be providing students with free access to our textbooks. Yes, the free Web format is going away, but our mission to provide high quality course materials at affordable prices remains as strong as ever. Students can read a complete online textbook with our Study Pass product, which includes note-taking, highlighting and study aids, for only $19.95. Our prices remain significantly lower than the $100+ that students are used to paying for other commercial textbooks."

So it seems it is up to individual academics (like us) to get quality study material directly to students for free and accept the fairly  modest compensation for paperback versions that on-demand publishers like CreateSpace offer.  

Any thoughts on this?

Saturday, May 19, 2012

Amazon Customers in Europe

press release
May 17, 2012, 9:00 a.m. EDT

CreateSpace Now Offers Independent Authors and Publishers Zero-Cost, Inventory-Free Distribution to Amazon Customers in Europe

Starting today, independent authors and publishers using CreateSpace can distribute their books directly to,,, and and earn industry-leading royalties...


Our textbooks are now available through Amazon in the UK, Germany, France, Italy, and Spain.  If you search on your country's Amazon under books using "real analysis" or "Bruckner and Thomson" you should be able to find us (soon?).   If you happen to purchase one of our textbooks this way please send us a note.   Sorry, but english language versions only are available at the moment.

As always, of course, free PDF files for all of our books are available at the web site.

Wednesday, May 9, 2012

Chinese Edition of Elementary Real Analysis

We are in the early stages of planning a Chinese edition of our elementary real analysis textbook.   If there is any interest out there, please write us words of encouragement or advice.

Sunday, May 6, 2012

We have now made the transition for our web site from M/S Office Live to another server.   You can access either of our sites:

for information about our real analysis texts, for free PDF downloads, for purchase of paperback editions, or for supplementary material [a work-in-progress] that will assist in your study or teaching of real analysis.

If you have any comments or advice on making these sites more useful please add your thoughts on this blog.

Friday, February 10, 2012

New web site


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:

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

Come back to

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