# Limit superior and inferior reference request.

Is there any book where I can find theory on the limit superior and limit inferior, plus a nice deal of excercises in the same spirit as Spivak's type of excercises?

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If it is exercises that you're looking for:

I would suggest that you look into the three volume book: Problems in Mathematical Analysis, by W J Kaczor and M T Nowak, Marie Curie-Sklodowska University, Lublin, Poland . This is published by the American Mathematical Society.

• The first volume deals with real numbers; sequences and series. This book is a very ideal start into a first course in Analysis. Amazing collection of problems. There are a lot of challenges. Some of the well known (read: hard) problems are given as exercises here. But, the necessay machinery is built by supplementing this with small toy problems. I learnt a lot from this book.

Randomly Chosen Problems from Vol. I:

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• The second volume deals with Continuity and Differentiability of Real-valued functions. The emphasis on semi-continuous functions; functions of Dirichlet's type are amazing. Some nice exercises as applications of Mean Value Theorem. There are lot of nice ideas in this book.

Randomly Chosen Problems from Vol. II:

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• The third volume covers the theory of Riemann and Lebesgue integrals. The collection of problems is richer than what is seen in any text book. They are also arranged beautifully.

Random Page from Vol. III: (Courtesy: the user t.b.)

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There are solutions that give hints--just about what is required to complete the problems. (I would view the solutions as complete; but it refers to previous exercises and you'll end up tracing a chain if you don't take the hints that are there.)

P.S. (The Indian Editions costs Rs. 800 each volume. Looking hard enough will get you these books from somewhere...)

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I don't get you PS, or do I? –  Pedro Tamaroff Apr 18 '12 at 20:31
I think you do. ;) –  user21436 Apr 18 '12 at 20:35

The first 2 volumes of Kaczor/Nowak have a large number of such exercises scattered throughout. Also, McShane's classic book on Lebesgue integration gives a well written and leisurely treatment. In the U.S., at least, you can find McShane's text in most every college and university library.

Kaczor/Nowak, Problems in Mathematical Analysis, Volumes 1 & 2, American Mathematical Society.

Edward James McShane, Integration, Princeton Mathematical Series, 1944, viii + 394 pages. [My copy is the 1974 8th printing.]

(added next day) After looking at McShane's book, I agree with the comments I wrote yesterday. It gives a very well written discussion of $\limsup$ and $\liminf$ for functions in Article 6 (pp. 26-38). Also, Article 7 (pp. 38-44) is nicely done. Below are 3 additional books that you might find useful (in light of your comment "Spivak's type of excercises"), but which don't specifically devote all that much to $\limsup$ and $\liminf$ for functions (at least, not together in one section).

Ralph Philip Boas, A Primer of Real Functions, 4th edition prepared by Harold Philip Boas, Mathematical Association of America, 1996, xiv + 305 pages.

Andrew Michael Bruckner, Judith Brostoff Bruckner, and Brian S. Thomson, Elementary Real Analysis, Prentice-Hall, 2001, xvi + 677 + 58 pages. [The final 58 pages consist of appendixes: (A) Background, (B) Hints for Selected Exercises, (C) Subject Index.]

Edgar Terome Townsend, Functions of Real Variables, Henry Holt and Company, 1928, xii + 405 pages.

Finally, more advanced readers of this thread may be interested in the following paper, which gives (on pp. 429-430) some relations involving mixed iterations of the operations $\limsup$ and $\liminf$ of a function at a point.

Robert Palmer Dilworth, The normal completion of the lattice of continuous functions, Transactions of the American Mathematical Society 68 (1950), 427-438.

http://www.ams.org/journals/tran/1950-068-03/S0002-9947-1950-0034822-9/

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Ah, I see Kannappan Sampath also recommended Kaczor/Nowak's books while I was preparing my reply. I really think they're exactly what you (Peter T.off) are looking for, but if a nearby library has McShane's book, you might want to look at it also. –  Dave L. Renfro Apr 18 '12 at 19:06
Is Edward's book's title "Unified Integration"? –  user21436 Apr 18 '12 at 19:53
@Kannappan Sampath: No, Unified Integration is a much later book (about 40 years later) that deals with the generalized Riemann integral (Henstock–Kurzweil integral). McShane's 1947 book was written for first year graduate students in real analysis (late 1940s), but much of the material is presented at a relatively leisurely pace (by today's standards), and it includes a lot of introductory and mostly self-contained background material (which includes the $\limsup$ and $\liminf$ topics). –  Dave L. Renfro Apr 18 '12 at 20:08
Thank You for telling me. I never knew it. I will search harder. :) –  user21436 Apr 18 '12 at 20:14
@Kannappan Sampath: I'll look through my copy of McShane's book when I get home tonight to make sure that my comments about its treatment of $\limsup$ and $\liminf$ are on the mark. Also, I'll look around and see if I can find other possibilities to recommend Peter T.off besides Kaczor/Nowak's books. –  Dave L. Renfro Apr 18 '12 at 20:16

Well,any good analysis text will have a discussion of it,Peter. I personally like the discussion in Angus Taylor's General Theory Of Functions And Integration, which is in Dover and has a lot of excellent exercises. But really,any good analysis text will do.

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Well, I couldn't find it in Spivak's or Apostol's, that's why I'm asking. Thanks for the reference. –  Pedro Tamaroff Apr 18 '12 at 18:35
@Peter Have you tried Apostol's MATHEMATICAL ANALYSIS rather then his calculus text? This is really a more sophisticated concept then you'll usually find in any calculus text,even advanced ones. –  Mathemagician1234 Apr 18 '12 at 18:52
Not really, I will! (*than) –  Pedro Tamaroff Apr 18 '12 at 18:56