From: Fred W. Helenius <fredh@ix.netcom.com>
Subject: Re: I'm looking for a good real analysis webpage.
Date: Thu, 27 Jul 2000 01:49:47 -0400
Newsgroups: sci.math
Summary: [missing]

Angelaruth49@webtv.net (Angela Olson) wrote:

>I am looking for a good real analysis webpage.  I took it in college
>and did poorly in it, so I would like to read it and see if I can find
>a glimmmer of understanding of it. 

There's a two-semester undergraduate course called Analysis WebNotes
available at http://www.math.unl.edu/~webnotes/home/home.htm .
==============================================================================

From: dlrenfro@gateway.net (Dave L. Renfro)
Subject: Re: I'm looking for a good real analysis webpage.
Date: 27 Jul 2000 13:29:39 -0400
Newsgroups: sci.math

Angela Olson <Angelaruth49@webtv.net>
[sci.math 26 Jul 00 22:50:14 -0400 (EDT)]
<http://forum.swarthmore.edu/epigone/sci.math/grersterdwee>

wrote

> I am looking for a good real analysis webpage.  I took it in
> college and did poorly in it, so I would like to read it and
> see if I can find a glimmmer of understanding of it. 
> 
> Have a nice evening,
> Angela

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I. REAL ANALYSIS LECTURE NOTES, TESTS, HANDOUTS, ETC.

1. John Lindsay Orr's Analysis WebNotes [Univ. of Nebraska-Lincoln]

<http://www.math.unl.edu/~webnotes/home/home.htm>

2. Bert G. Wachsmuth's Interactive Real Analysis [Seton Hall Univ.]

<http://www.shu.edu/projects/reals/cont/index.html>

3. Ian Craw's text for MA1002: Advanced Calculus and Analysis

<http://www.maths.abdn.ac.uk/~igc/tch/ma2001/notes/notes.html>

4. Lee Larson's Real Analysis Lecture Notes [Univ. of Louisville]

<http://www.math.louisville.edu/~lee/RealAnalysis/realanalysis.html>

5. Curtis T. McMullen's notes for Real Analysis and Advanced Real
   Analysis [Harvard Univ.]

<http://www.math.harvard.edu/~ctm/past.html>

6. Gianluca Gorni's various handouts and analysis I, II notes
   (in Italian) [Univ. di Udine]

<http://www.dimi.uniud.it/~gorni/Dispense/index.html#Dispense>
<http://www.dimi.uniud.it/~gorni/Analisi1/index.html#Analisi 1>
<http://www.dimi.uniud.it/~gorni/Analisi2/index.html#Analisi 2>

7. Caltech's Math 108a,b,c: Classical Analysis

<http://www.math.caltech.edu/courses/99ma108a.html>
<http://www.math.caltech.edu/courses/00ma108b.html>
<http://www.math.caltech.edu/courses/00ma108c.html>

8. Joel Feinstein's handouts and lecture notes in real analysis
   [Univ. of Nottingham]

<http://www.maths.nott.ac.uk/personal/jff/G12RAN/>

9. MA203: Real Analysis (course notes and previous exams), at the
    London School of Economics

<http://www.maths.lse.ac.uk/Courses/MA203/index.html>
<http://www.maths.lse.ac.uk/Courses/ma203.html#exams>

10. MAT 3135; notes in French on Lebesgue integration (340 K .dvi,
    508 K .pdf, 712 K .ps) [Univ. de Montréal]

<http://www.dms.umontreal.ca/~giroux/mat3135.htm>

11. Dr. Vogel's Gallery of Calculus Pathologies [Texas A & M Univ.]

<http://www.math.tamu.edu/~tom.vogel/gallery/gallery.html>

12. THE CALCULUS PAGE PROBLEMS LIST by D. A. Kouba [Univ. of Calif.
    at Davis] The problems in the categories "precise epsilon/delta
    definition", "continuity of a function", " "Squeeze Principle" ",
    and "limit definition of the derivative" contain some problems
    (all with solutions) that are sufficiently sophisticated for
    an intermediate level undergraduate real analysis course.

<http://www.math.ucdavis.edu/~kouba/ProblemsList.html>

13. Maria Girardi's tests for Real Analysis (undergraduate),
    Analysis I and II (graduate) [Univ. of South Carolina]

<http://www.math.sc.edu/~girardi/w554.html>
<http://www.math.sc.edu/~girardi/w7034.html>

14. Noel Vaillant's Probability Tutorials (Extensive lecture notes
    on graduate level real analysis topics.)

<http://www.probability.net/>

15. International Baccalaureate: Higher Level Mathematics:
    Option 12. Analysis and Approximation

<http://is7.pacific.net.hk/~cismath/Anal&Approx.htm>

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II. SOME USEFUL COLLECTIONS OF LINKS

1. Dave Rusin's The Mathematical Atlas 26: Real functions and
   Dave Rusin's The Mathematical Atlas 28: Measure and integration.
   See especially "Selected topics at this site" at the bottom of
   each of these web pages.

<http://www.math.niu.edu/~rusin/known-math/index/26-XX.html>
<http://www.math.niu.edu/~rusin/known-math/index/28-XX.html>

2. Bruno Kevius' calculus and analysis links (a huge collection)

<http://www.abc.se/~m9847/matre/calculus.html>

3. Math Forum Internet Mathematics Library: Math Topics:
   Analysis: Real Analysis

<http://forum.swarthmore.edu/library/topics/real_a/>

4. The Math Forum: Ask Dr Math: Questions and Answers from Our
   Archives: Analysis AND Logic and Set Theory

<http://forum.swarthmore.edu/dr.math/tocs/analysis.college.html>
<http://forum.swarthmore.edu/dr.math/tocs/logic.college.html>

5. Chris Hillman's Math 424-5-6 Home Page [Univ. of Washington]

<http://www.math.washington.edu/~hillman/424.html>

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III. SOME OF MY INTERNET POSTS INVOLVING REAL ANALYSIS

     I'm only giving the entire thread's URL, not my
     specific post URL'S. This is because the URL for the
     specific posts tends to be very long.

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A. ELEMENTARY TOPICS

1. WHAT I DISCUSS: How to integrate (1 + x^4)^(-1).
   ACTUAL THREAD TITLE: how to integrate this function?
   DATE LISTING FOR MY POST: Oct. 19 and 22, 1999

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/quumwhendu>

2. WHAT I DISCUSS: A fairly complete survey of elementary ways to
                   evaluate limits.
   ACTUAL THREAD TITLE: [tan(sin x) - sin(tan x)]/(x^7) as x --> 0
   DATE LISTING FOR MY POST: Feb. 13, 2000 and April 16, 2000

<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul>

3. WHAT I DISCUSS: Complete text of Cayley's 1895 paper "On the
                   value of tan(sin x) - sin(tan x)".
   ACTUAL THREAD TITLE: [tan(sin x) - sin(tan x)]/(x^7) as x --> 0
   DATE LISTING FOR MY POST: July 11, 2000

<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul>

4. WHAT I DISCUSS: Complete details for the integration of
                   (1 + x - x^2)^(1/2).
   ACTUAL THREAD TITLE: Hard integrals have any?
   DATE LISTING FOR MY POST: Nov. 20, 1999

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/stahglimpchax>

5. WHAT I DISCUSS: A discussion of, and a specific worked example
                   of, Feynman's "differentiating under the integral
                   sign" method for evaluating integrals, along with
                   a quote from "Surely You're Joking Mr. Feynman"
                   where Feynman mentions this technique.
   ACTUAL THREAD TITLE: Feynman
   DATE LISTING FOR MY POST: April 8, 2000

<http://forum.swarthmore.edu/epigone/sci.math/wehelcrerm>

6. WHAT I DISCUSS: Issues related to x^x when x is negative.
   ACTUAL THREAD TITLE: Question on rational exponents
   DATE LISTING FOR MY POST: March 31, 2000 (A small correction
                             is given in my April 1 follow-up post.)

<http://forum.swarthmore.edu/epigone/math-teach/teldpahgeh>

7. WHAT I DISCUSS: Discussion of the limit comparison test and the
                   difficulties of determining series convergence
                   by numerical investigations.
   ACTUAL THREAD TITLE: convergence or divergence?
   DATE LISTING FOR MY POST: Nov. 30, 1999 (two posts)

<http://forum.swarthmore.edu/epigone/ap-calc/jexdweelor>

8. WHAT I DISCUSS: Evaluation of a limit using the Taylor series
                   method.
   ACTUAL THREAD TITLE: Limit Problem
   DATE LISTING FOR MY POST: Dec. 2, 1999

<http://forum.swarthmore.edu/epigone/ap_calc/cryphohoi> 

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B. INTERMEDIATE TOPICS

1. WHAT I DISCUSS: Results about the convergence of x^x^x^...
   ACTUAL THREAD TITLE: Problems in (0,1/e)
   DATE LISTING FOR MY POST: Sept. 9, 1999

<http://forum.swarthmore.edu/epigone/sci.math/ranbrelbrel>

2. WHAT I DISCUSS: Some additional references to the convergence
                   of x^x^x^...
   ACTUAL THREAD TITLE: polypowers/iterated exponentials
   DATE LISTING FOR MY POST: Dec. 26, 1999 (first post)

<http://forum.swarthmore.edu/epigone/sci.math.research/yendrendswil>

3. WHAT I DISCUSS: Several references for slowly diverging and
                   slowly converging series.
   ACTUAL THREAD TITLE: Convergent or Divergent series???
   DATE LISTING FOR MY POST: Sept. 10, 1999

<http://forum.swarthmore.edu/epigone/sci.math/paxzelcrex>

4. WHAT I DISCUSS: Two proofs that any closed set of real numbers
                   is equal to the set of subsequence limit points
                   of some sequence.
   ACTUAL THREAD TITLE: Types Of Limits
   DATE LISTING FOR MY POST: June 28, 2000

<http://forum.swarthmore.edu/epigone/sci.math/zhimpprudwel>

5. WHAT I DISCUSS: Proof that the maximal ideals in the ring of
                   real-valued continuous functions is the
                   collection of zero sets of singletons.
   ACTUAL THREAD TITLE: Ring Theory
   DATE LISTING FOR MY POST: Aug. 23, 1999

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/zhahchumsmal>

6. WHAT I DISCUSS: Some web pages having examples of mathematical
                   induction proofs.
   ACTUAL THREAD TITLE: induction
   DATE LISTING FOR MY POST: Sept. 11, 1999

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/sparfloxspix>

7. WHAT I DISCUSS: Outline of a proof that if f is continuous on
                   [a,b] with  a maximum value of M, then
                   limit(n --> infinity) of the integral from a to b
                   of [f(x)]^n equals M.
   ACTUAL THREAD TITLE: limit of an integral
   DATE LISTING FOR MY POST: Feb. 28, 2000

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/shumtwabu>

8. WHAT I DISCUSS: Strictly increasing continuous functions.
   ACTUAL THREAD TITLE: Real Analysis Q
   DATE LISTING FOR MY POST: March 14, 2000

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/yillundwax>

9. WHAT I DISCUSS: Web pages where nowhere differentiable continuous
                   functions are exhibited and/or discussed.
   ACTUAL THREAD TITLE: Fonction réelle
   DATE LISTING FOR MY POST: April 26, 2000

<http://forum.swarthmore.edu/epigone/sci.math/jogruspo>

10. WHAT I DISCUSS: Excerpt from R. P. Agnew's differential equation
                    text about some subtilies involved in factoring
                    differential equations. The discussion involves
                    an application of Rolle's theorem.
    ACTUAL THREAD TITLE: breaks in domain (was RE: log cabin, song,
                         & competition)
    DATE LISTING FOR MY POST: Dec. 16, 1999

<http://forum.swarthmore.edu/epigone/ap-calc/sixflarten>

11. WHAT I DISCUSS: I give a proof that every derivative satisfies
                    the intermediate value property.
    ACTUAL THREAD TITLE: Why is an integral the antiderivative?
    DATE LISTING FOR MY POST: March 18, 2000

<http://forum.swarthmore.edu/epigone/sci.math/zheldkheetwar>

12. WHAT I DISCUSS: Discussion, with several web pages given, of
                    continuity matters--especially of the ruler
                    function. I also give a proof that the ruler
                    function is continuous at each irrational point
                    and discontinuous at each rational point.
    ACTUAL THREAD TITLE: Continuity on open, closed intervals
    DATE LISTING FOR MY POST: Nov. 14, 1999

<http://forum.swarthmore.edu/epigone/ap-calc/daxkookay>

13. WHAT I DISCUSS: A detailed discussion of various notions of
                    "increasing at a point" vs "increasing on an
                    interval" vs "having a positive derivative".
    ACTUAL THREAD TITLE: question on boundaries
    DATE LISTING FOR MY POST: April 25, 2000

<http://forum.swarthmore.edu/epigone/ap-calc/rulmaybrox>

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C. ADVANCED TOPICS

1. WHAT I DISCUSS: Mathematical and historical issues concerning
                   discontinuity points of derivatives.
   ACTUAL THREAD TITLE: HISTORICAL ESSAY ON CONTINUITY OF
                        DERIVATIVES
   DATE LISTING FOR MY POST: Jan. 23, 2000

<http://forum.swarthmore.edu/epigone/sci.math/pringquibroi>

2. WHAT I DISCUSS: A detailed survey of the fact that countable
                   unions of closed Lebesgue null sets generate
                   a sigma-ideal properly contained in the
                   sigma-ideal of sets simultaneously Lebesgue null
                   and first category, along with the observation
                   that the exceptional set for the Riemann
                   integrability continuity condition is a countable
                   union of closed Lebesgue null sets.
   ACTUAL THREAD TITLE:  HISTORICAL ESSAY ON F_SIGMA LEBESGUE
                         NULL SETS
   DATE LISTING FOR MY POST: May 1, 2000

<http://forum.swarthmore.edu/epigone/sci.math/dwillydwul>

3. WHAT I DISCUSS: Essay on the set of points of non-uniform
                   convergence of point-wise convergent sequences
                   of functions.
   ACTUAL THREAD TITLE: Continuity of Fourier
   DATE LISTING FOR MY POST: May 20, 2000

<http://forum.swarthmore.du/epigone/sci.math/frorchestox>

4. WHAT I DISCUSS: Some comments on the origin of the phrase
                   "almost everywhere".
   ACTUAL THREAD TITLE: [HM] Presque partout
   DATE LISTING FOR MY POST: June 25, 26 and July 26, 2000

<http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox>

5. WHAT I DISCUSS: Two definitions of "approximate limit", along
                   with some elementary remarks on approximate
                   continuity and approximate differentiation.
   ACTUAL THREAD TITLE: Lesbegue measure etc. - help, plz.
   DATE LISTING FOR MY POST: June 27, 2000

<http://forum.swarthmore.edu/epigone/sci.math/snugrepru>

6. WHAT I DISCUSS: Some discussions on "points of non-measurability"
                   of a set. (TWO threads on this topic.)
   ACTUAL THREAD TITLE: Points of non-measurability
   DATE LISTING FOR MY POST: Nov. 25, 28, 29, 1999

<http://forum.swarthmore.edu/epigone/sci.math/nenvochoi>

   ACTUAL THREAD TITLE: [HM] Italian to English requested
   DATE LISTING FOR MY POST: July 21, 2000

<http://forum.swarthmore.edu/epigone/historia_matematica/zhangjorprim>

7. WHAT I DISCUSS: Some historical remarks and references for
                   Du Bois-Reymond's and Cantor's method of
                   condensation of singularities.
   ACTUAL THREAD TITLE: funky function
   DATE LISTING FOR MY POST: Sept. 4, 1999

<http://forum.swarthmore.edu/epigone/sci.math/quinnoxlel>

8. WHAT I DISCUSS: A survey, with references, to some results about
                   bases for the real numbers as a vector space over
                   the rational numbers (primarily Lebesgue measure
                   and Baire category results).
   ACTUAL THREAD TITLE: Hamel bases (was: Re: PI...)
   DATE LISTING FOR MY POST: Oct. 2, 1999

<http://forum.swarthmore.edu/epigone/sci.math/kahcrugror>

9. WHAT I DISCUSS: Discussion of discontinuous solutions
                   to f(x+y) = f(x) + f(y)
   ACTUAL THREAD TITLE: functional equations
   DATE LISTING FOR MY POST: Aug. 14, 1999

<http://forum.swarthmore.edu/epigone/k12.ed.math/swintendring>

10. WHAT I DISCUSS: A survey of results concerning the set of points
                    at which a measurable function has a symmetric
                    derivative but not an ordinary derivative.
    ACTUAL THREAD TITLE: Help: symmetric derivative
    DATE LISTING FOR MY POST: Oct. 3, 1999

<http://forum.swarthmore.edu/epigone/sci.math/bandglixkerm>

11. WHAT I DISCUSS: Some fractal dimension, and other, results
                    concerning Cantor sets construced by a dyadic
                    process.
    ACTUAL THREAD TITLE: newbie question: Cantor generalization
    DATE LISTING FOR MY POST: Dec. 15, 1999

<http://forum.swarthmore.edu/epigone/sci.math/thaxprenvil>

12. WHAT I DISCUSS: Remarks on Scheeffer's theorem (1884): Given a
                    perfect nowhere dense set C and a countable
                    set Z, let B be the set of numbers b such that
                    the b-translate of C is disjoint from Z. Then B
                    is dense in the reals.
    ACTUAL THREAD TITLE: Question from Rudin
    DATE LISTING FOR MY POST: May 11, 2000

<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal>

13. WHAT I DISCUSS: Lengthy discussion of the Cantor-Bendixson rank
                    of a closed set, the early history of
                    trigonometric sets of uniqueness, and a proof
                    that every countable linearly ordered set is
                    order-isomorphic to a subset of the rationals.
    ACTUAL THREAD TITLE: Question from Rudin
    DATE LISTING FOR MY POST: May 15, 2000

<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal>

14. WHAT I DISCUSS: Some comments on Kakeya sets and the Kakeya
                    Conjecture, some of the more important published
                    papers in the field, and an extensive list of
                    references available on the internet.
    ACTUAL THREAD TITLE: Worms
    DATE LISTING FOR MY POST: May 18, 2000

<http://forum.swarthmore.edu/epigone/sci.math/blangchimsmen>

15. WHAT I DISCUSS: Remarks on the density of nowhere differentiable
                    continuous functions, and functions whose graphs
                    have a specified Hausdorff dimension, in the
                    space of continuous functions with the sup norm.
    ACTUAL THREAD TITLE: Hausdorff distance and fractal dimension
    DATE LISTING FOR MY POST: Jan. 19, 2000

<http://forum.swarthmore.edu/epigone/sci.math.research/smarclurerd>

16. WHAT I DISCUSS: Nearly 40 links to old Ph.D. math qualifying
                    exams at various universities. Virtually all
                    of these links have problems involving real
                    analysis and many of these are provided with
                    solutions.
    ACTUAL THREAD TITLE: WEB PAGES FOR PH.D. QUALIFYING EXAMS
    DATE LISTING FOR MY POST: June 23, 2000

<http://forum.swarthmore.edu/epigone/sci.math.research/mermswaplou>

17. WHAT I DISCUSS: Some internet sites dealing with Apollonian
                    packings.
    ACTUAL THREAD TITLE: Curved plane figure
    DATE LISTING FOR MY POST: Feb. 4, 2000

<http://forum.swarthmore.edu/epigone/math-teach/zhugarsweld>

18. WHAT I DISCUSS: Example of a function satisfying the intermediate
                    value property that is not Riemann integrable
                    over any closed subinterval of [0,1].
    ACTUAL THREAD TITLE: [MATHEDU] IVProperty
    DATE LISTING FOR MY POST: Jan. 18, 2000. My post is contained
                              in David Epstein's Jan. 18, 2000 post.
                              Epstein's inserted comment about one of
                              my remarks is true, but not relevant
                              since the empty set is a G_delta set.
                              (Also, my comment that the points of
                              continuity form a residual set is only
                              made for the specific function at
                              hand.)

<http://forum.swarthmore.edu/epigone/mathedu/chorskihun>

19. WHAT I DISCUSS: Some remarks about pathological functions having
                    the intermediate value property are given in my
                    comments to the original poster's 2'nd question.
    ACTUAL THREAD TITLE: real analysis
    DATE LISTING FOR MY POST: Nov. 25, 1999

<http://forum.swarthmore.edu/epigone/alt.math.undergrad/kayglanwhel>

20. WHAT I DISCUSS: A list of references for the use of nets in
                    analysis, including a lot of information about
                    Alan F. Beardon's book "LIMITS: A New Approach
                    to Real Analysis".
    ACTUAL THREAD TITLE: nets (generalized sequences) in analysis
    DATE LISTING FOR MY POST: April 18, 2000

<http://forum.swarthmore.edu/epigone/sci.math/hilrulwa>

21. WHAT I DISCUSS: A list of references concerning Haar null sets
                    and a summary of some results in Hongjian Shi's
                    Ph.D. 1997 Ph.D. Dissertation (under Brian S.
                    Thomson) "Measure-Theoretic Notions of
                    Prevalence".
    ACTUAL THREAD TITLE: "measure zero" on infinite dimensional
                         spaces
    DATE LISTING FOR MY POST: Oct. 17 and 18, 1999

<http://forum.swarthmore.edu/epigone/sci.math.research/cleekhenbrum>

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Dave L. Renfro