WEB PAGES FOR REAL ANALYSIS
Dave L. Renfro
lecture notes, handouts, tests, etc. on real analysis that I posted
in the sci.math thread "I'm looking for a good real analysis
webpage." on July 27, 2000 at
<http://forum.swarthmore.edu/epigone/sci.math/grersterdwee>.
I. REAL ANALYSIS LECTURE NOTES, TESTS, HANDOUTS, ETC. (30 items)
II. SOME USEFUL COLLECTIONS OF LINKS (11 items)
III. SOME OF MY INTERNET POSTS INVOLVING REAL ANALYSIS
A. ELEMENTARY TOPICS, INCLUDING CALCULUS (20 items)
B. ADVANCED UNDERGRADUATE TOPICS (33 items)
C. GRADUATE LEVEL TOPICS (35 items)
Dave L. Renfro
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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, Italy]
<http://www.dimi.uniud.it/~gorni/Dispense/index.html#Dispense>
<http://www.dimi.uniud.it/~gorni/Analisi1/index.html#Analisi>
<http://www.dimi.uniud.it/~gorni/Analisi2/index.html#Analisi>
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 (a) real analysis
and (b) metric and topological spaces [Univ. of Nottingham]
<http://www.maths.nott.ac.uk/personal/jff/G12RAN/>
<http://www.maths.nott.ac.uk/personal/jff/G13MTS/index.html>
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 and problem solutions in French on undergraduate
real analysis (509 .pdf file for the notes) and for Lebesgue
integration (561 K .pdf file for the notes) [Univ. de Montréal]
<http://www.dms.umontreal.ca/~giroux/analyse_2.htm>
<http://www.dms.umontreal.ca/~giroux/mesure.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.)
15. Chris Hillman's notes "What is Hausdorff Dimension?" (331 K .ps
file, 12 page output, for 1995 version; I believe an expanded
version is being prepared.)
<http://www.math.washington.edu/~hillman/PUB/newhd.ps>
16. John Shackell's notes for MA 571 Real Analysis and Metric Spaces
(Univ. of Kent at Canterbury) ["Notes on subsequences" (114 K .ps
file, 5 page output); "Notes on Riemann integration" (132 K .ps
file, 12 page output); "Notes on metric spaces" (207 K .ps file,
16 page output); "Notes on uniform convergence" (146 K .ps file,
10 page output)]
<http://www.ukc.ac.uk/IMS/maths/people/J.R.Shackell/MA571/ma571.html>
17. Timothy Gowers "Mathematical discussions contents page". [Topics
under "Analysis" include: "A dialogue concerning the existence of
the square root of two"; "The meaning of continuity"; "How to
solve basic analysis exercises without thinking"; Proving that
continuous functions on the closed interval [0,1] are bounded";
"Finding the basic idea of a proof of the fundamental theorem of
algebra"; "What is the point of the mean value theorem?"; "A tiny
remark about the Cauchy-Schwarz inequality"]
<http://www.dpmms.cam.ac.uk/~wtg10/mathsindex.html>
18. Michael Filaseta's notes for Math 555: Real Analysis II [Univ.
of South Carolina]
<http://www.math.sc.edu/~filaseta/courses/Math555/Math555.html>
19. Kenneth Kuttler's notes for math 541: real analysis [Brigham
Young Univ.] (3896 K .ps file, 538 page output) [In addition to
the standard topics in a graduate real analysis class, these
notes include chapters titled: "Fourier series", "The Frechet
derivative", "Change of variables for C^1 maps", "Fourier
transforms" (includes sections on distributions), "Brouwer
degree", "Differential forms", "Lipschitz manifolds", and "The
generalized Riemann integral".]
<http://www.math.byu.edu/~klkuttle/math541notes.ps>
20. Math 105A: Real Analysis [Univ. of California at Santa Cruz]
<http://orca.ucsc.edu/math105a/contents.html>
<http://orca.ucsc.edu/math105a/old_notes.html>
21. Yuri Safarov's notes for Real Analysis CM 321A [King's College
London]
<http://www.mth.kcl.ac.uk/~ysafarov/Lectures/CM321A/>
22. Karen E. Donnelly's notes for M345 Real Analysis [Fall 2000 at
Saint Joseph's College]
<http://www.saintjoe.edu/~karend/m345/>
23. Erhan Cinlar and Robert J. Vanderbei's book "Real Analysis for
Engineers" (483 K .pdf file, 119 page output)
<http://www.princeton.edu/~rvdb/506book/book.pdf>
24. Vitali Liskevich's [Univ. of Bristol] "Lecture Notes on Measure
Theory and Integration" (97 K .dvi file, 40 page output) AND
notes for Analysis - 1 (331 K .dvi file; 1283 K .ps file,
137 page output)
<http://www.maths.bris.ac.uk/~pure/staff/maval/c98.dvi>
<http://www.maths.bris.ac.uk/~pure/staff/maval/an1.html>
25. Notes for MATH 2610 HIGHER REAL ANALYSIS - Session 1, 2001
at Univ. of New South Wales
<http://www.maths.unsw.edu.au/ForStudents/courses/math2610/notes.html>
26. James H. Money's "Notes for analysis" ["Definite Integral" (111 K
.pdf file; "Cauchy Sequences" (126 K .pdf file); "LUB Proof"
(66 K .pdf file)]
<http://www.math.jmu.edu/~jmoney/analysis/>
27. Stephen Lich-Tyler's "Who wants to be a mathematical economist?"
[Univ. of Texas at Austin] (See: 8. Analysis: basics of real
analysis; 51 K .pdf file, 7 page output)
<http://www.eco.utexas.edu/graduate/Lich-Tyler/math2/>
28. Jim Langley's notes for real analysis [Univ. of Nottingham]
(393 K .pdf file)
<http://www.maths.nott.ac.uk/personal/jff/G12RAN/pdf/JKL.pdf>
29. Joel H. Shapiro's Lecture Notes [Michigan State Univ.]
(Includes: "Notes on the dynamics of linear operators"; "The
Arzela-Ascoli Theorem"; "Notes on Differentiation"; "A Gentle
Introduction to Composition Operators"; "Nonmeasurable
sets and paradoxical decompositions")
<http://www.mth.msu.edu/~shapiro/Pubvit/LecNotes.html>
30. Leif Abrahamsson's course material for real analysis
[Uppsala Univ., Sweden]
<http://www.math.uu.se/~leifab/Reellanalys/annat/kursmat00h.html>
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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. WEB PAGES FOR PH.D. QUALIFYING EXAMS (See the latest version,
which at present is dated June 23, 2000.) [Virtually all of these
contain a number of tests (often with solutions) in graduate
level real analysis.]
<http://forum.swarthmore.edu/epigone/sci.math/howousway>
6. STUDENT SEMINAR AND SENIOR CAPSTONE REFERENCES [Some of the items
listed in Section V: "papers, projects, essays by or for students"
involve real analysis topics.]
<http://forum.swarthmore.edu/epigone/sci.math/gosmangzha/>
7. Chris Hillman and Roland Gunesch's resources for Entropy in
Ergodic Theory and Dynamical Systems
<http://www.math.psu.edu/gunesch/Entropy/dynsys.html>
8. Terence Tao's "Harmonic Analysis Links"
<http://www.math.ucla.edu/~tao/harmonic.html>
9. Online Books and Lecture Notes in Mathematics
<http://www.gotmath.com/notes.html>
<http://www.wam.umd.edu/~mcmahonj/notes.html>
<http://dmoz.org/Science/Math/Publications/Online_Texts/>
10. IB Higher Level Mathematics: Option 12. Analysis and
Approximation [A useful list of many topics that arise in
a honors calculus course or a in lower level real analysis
course, with links to web pages for more about the topics.]
<http://www.cis.edu.hk/sec/math/Anal&Approx.htm>
11. Michael Botsko's [Saint Vincent College] "A Web Page in Real
Analysis"
<http://facweb.stvincent.edu/academics/mathematics/Ranal/pub.html>
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III. SOME OF MY INTERNET POSTS INVOLVING REAL ANALYSIS
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A. ELEMENTARY TOPICS [Roughly arranged this way: Precalculus topics,
differentiation, integration, sequences and
series, ODE's, partial differentiation.]
1. Some really complicated surd (i.e. radical) simplifications.
<http://forum.swarthmore.edu/epigone/sci.math/foinulstix/klea64...@forum.swarthmore.edu>
2. 9 applications of rationalizing the numerator or denominator.
[See the June 13, 2001 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/mehspyswimp>
3. Why we don't prove a trig. identity as if we were solving an
equation. [Correction given in 2'nd URL.]
<http://forum.swarthmore.edu/epigone/sci.math/chimpleekax/juj2b2...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/chimpleekax/5wwhmx...@forum.mathforum.com>
4. Why don't we use negative bases for exponential functions?
<http://forum.swarthmore.edu/epigone/math-teach/salcrunstou/3w47rj...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/math-teach/salcrunstou/arjlxe...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/whenspedy/2h2q22...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/whenspedy/js6r0y...@forum.mathforum.com>
5. Remarks about Carl Sagan's novel "Contact", messages in Pi, and
normal numbers.
<http://forum.swarthmore.edu/epigone/sci.math/glangquingzhal/xkhzfk...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/math-teach/wingdwimpfrang/xld0w5...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/math-teach/wingdwimpfrang/l8hw4v...@forum.mathforum.com>
6. Some remarks about the use of the intermediate value property of
continuous functions on intervals for solving inequalities and
8 URL's for worked calculus curve sketching examples. [See the
May 6, 2001 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/grithanggix>
7. Some examples where local max/min's of [f(x)]^[g(x)] can be
explicitly found.
<http://forum.swarthmore.edu/epigone/sci.math/whenspedy/ni94df...@forum.mathforum.com>
8. Two definitions of an asymptote -- (i) a line the graph approaches
at infinity; (ii) a line that both the graph and the slope of the
graph approaches at infinity. [See the Sept. 21, 2000 post by
Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/snedikha>
9. A discussion of L'Hopital's rule and several web pages about the
continuity and differentiability of x*sin(1/x) and (x^2)*sin(1/x).
[See the Sept. 24 and 25, 2000 posts by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/grisworleld>
10. A graphical look at a discontinuous derivative and several links
to other posts of mine that deal with properties of derivatives.
[See the Dec. 1, 2000 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/wigufren>
11. Complete details for the integration of (1 + x - x^2)^(1/2).
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/stahglimpchax/040vtk...@forum.swarthmore.edu>
12. How to integrate (1 + x^4)^(-1). [Correction given in 2'nd URL.]
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/quumwhendu/zpmaae...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/quumwhendu/1fx0rr...@forum.swarthmore.edu>
13. Feynman's method of differentiating under the integral sign and
an excerpt from "Surely You're Joking, Mr. Feynman" where this
method is mentioned.
<http://forum.swarthmore.edu/epigone/sci.math/wehelcrerm/j0yygw...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/wehelcrerm/6htqju...@forum.mathforum.com>
14. How to evaluate integrals from -infinity to +infinity of
(polynomial)*exp(-a*x^2 + b*x + c) for a > 0.
<http://forum.swarthmore.edu/epigone/sci.math/swahstimpquer/2svesu...@forum.mathforum.com>
15. Comments and references about series that converge or diverge
very slowly.
<http://forum.swarthmore.edu/epigone/sci.math/pendsnenpun/j51fl2...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/pendsnenpun/8uz9my...@forum.mathforum.com>
16. The Taylor series method for evaluating limits.
<http://forum.swarthmore.edu/epigone/ap_calc/cryphohoi/so22ne...@forum.swarthmore.edu>
17. A survey of a lot of elementary ways to evaluate limits.
<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/3lmtwp...@forum.swarthmore.edu>
18. Concerning the limit [tan(sin x) - sin(tan x)]/(x^7) as x --> 0.
<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/at3or5...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/mnb2im...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/friblorkhul/9eueva...@forum.mathforum.com>
19. Excerpt from R. P. Agnew's differential equation text about some
subtleties involved in factoring differential equations. The
discussion involves an application of Rolle's theorem.
<http://forum.swarthmore.edu/epigone/ap-calc/sixflarten/1js7yq...@forum.swarthmore.edu>
20. How to prove 1=2 by partial differentiation.
<http://forum.swarthmore.edu/epigone/sci.math/zheldzermsna/xkpcu7...@forum.mathforum.com>
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B. INTERMEDIATE TOPICS [Roughly arranged this way: logic and set
theory, number theory related, metric and
topological properties of subsets of R^n,
continuity and differentiability behavior
of functions, integrability issues.]
1. Three proofs that a set with n elements has 2^n subsets.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/neesnimpyim/catnz4...@forum.mathforum.com>
2. Some web pages having examples of mathematical induction proofs.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/sparfloxspix/9fo7r8...@forum.swarthmore.edu>
3. Some REALLY large numbers. [The 4'th URL gives some remarks by
David Libert (June 14 and 15, 2001) on the Howard ordinal. The
2'nd group of URL's is a lengthy essay by Robert Munafo.]
<http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/8ougi2...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/b2p89z...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/gunsnersteh/l8168b...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/shingvixtwimp>
<http://home.earthlink.net/~mrob/pub/math/largenum.html>
<http://home.earthlink.net/~mrob/pub/math/largenum-2.html>
<http://home.earthlink.net/~mrob/pub/math/largenum-3.html>
<http://home.earthlink.net/~mrob/pub/math/largenum-4.html>
4. A list of 16 web pages dealing with elementary aspects of the
cardinality of sets, along with my comments on them.
<http://forum.swarthmore.edu/epigone/sci.math/clerdmandprim/7flq8i...@forum.mathforum.com>
5. Some texts on elementary set theory and my comments on them.
<http://forum.swarthmore.edu/epigone/sci.math/crahseenee/i3406y...@forum.swarthmore.edu>
6. An essay on how far the transfinite sequence of cardinal numbers
extends, going well past the first ordinal b such that b = Beth_b.
<http://forum.swarthmore.edu/epigone/sci.math/clerdmandprim/h6yjoc...@forum.mathforum.com>
7. Some references for Cohen's CH independence result for someone
just beginning (plus some related internet sites).
<http://forum.swarthmore.edu/epigone/sci.math/zexglysnoo/3uwd7v...@forum.swarthmore.edu>
8. Proofs that there are infinitely many primes using relatively
advanced mathematical ideas.
<http://forum.swarthmore.edu/epigone/sci.math/thoxquankee>
9. References for the prime number theorem and the divergence of
the harmonic prime series.
<http://forum.swarthmore.edu/epigone/sci.math/ryblilfral/37E88457...@gateway.net>
<http://forum.swarthmore.edu/epigone/ap-calc/stelthenddwel/s3sn0p...@forum.mathforum.com>
10. Extensive list of references (print and internet) for proofs
that (1) e is irrational, (2) Pi is irrational, (3) e is
transcendental, and (4) Pi is transcendental.
<http://forum.swarthmore.edu/epigone/sci.math/zhingpheldkoi/zmek2c...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/zhingpheldkoi/5jitzz...@forum.mathforum.com>
11. Proof that a trig. function of a rational number of degrees is
an algebraic number, along with several literature references.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/vogoubril/27t25c...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/vogoubril/cq3ssi...@forum.swarthmore.edu>
12. Brief discussion of whether SIN(1 deg) is (a) a constructible
number (in the ruler and compass sense), (b) an explicit
algebraic number, and/or (c) an algebraic number. Explicit
expressions (using radicals and rational numbers) for the exact
values for SIN(3 deg) and SIN(180/17 deg) are given.
<http://forum.swarthmore.edu/epigone/math-teach/sheltolla/ikd4eb...@forum.mathforum.com>
13. Remarks about explicit algebraic numbers and algebraic numbers.
<http://forum.swarthmore.edu/epigone/sci.math/poxcrimpvo/2p0zdw...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/pqd9rd...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/dw03ob...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/kf2fw9...@forum.mathforum.com>
14. Two proofs that any closed set of real numbers is equal to the
set of cluster points of some sequence.
<http://forum.swarthmore.edu/epigone/sci.math/zhimpprudwel/t09kob...@forum.mathforum.com>
15. Some fractal dimension, and other, results concerning
Cantor sets constructed by a dyadic process.
<http://forum.swarthmore.edu/epigone/sci.math/thaxprenvil/97lr7e...@forum.swarthmore.edu>
16. Some references and historical remarks about a result that
Cantor proved in 1882 -- If D is a countable subset of R^n,
then R^n - D is pathwise connected.
<http://forum.swarthmore.edu/epigone/sci.math/blexspehwhin/q4du0d...@forum.mathforum.com>
17. Discussion about and web page references to continuity
matters--especially of the ruler function. A proof is given
that the ruler function is continuous at each irrational point
and discontinuous at each rational point.
<http://forum.swarthmore.edu/epigone/ap-calc/daxkookay/382D63E9...@gateway.net>
18. A discussion of various notions of "increasing at a point" vs.
"increasing on an interval" vs. "having a positive derivative".
<http://forum.swarthmore.edu/epigone/ap-calc/rulmaybrox/ruq95j...@forum.mathforum.com>
19. Geometric discussion of locally linear and a generalization due
to Bouligand that applies to any subset of the plane (the
contingent of a set). [See Nov. 27, 2000 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap_calc/grandzaygren>
20. Examples of functions whose tangent line errors approach
zero slower than quadratically. [See Nov. 28, 2000 post by
Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap_calc/grandzaygren>
21. Essay on "differentiable ==> continuity", leading to a discussion
of point-wise modulus of continuity conditions and ending with
some remarks about the Wierstrass nowhere differentiable
function. [See Dec. 4, 2000 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/wigufren>
22. A survey of the relation between geometrical concavity
conditions and continuity/differentiability conditions.
<http://forum.swarthmore.edu/epigone/ap-calc/khendgeequer>
<http://forum.swarthmore.edu/epigone/ap-calc/stelzonzhel>
23. Some internet web pages for epsilon/delta proof examples
and discussions. [See the three posts by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap_calc/whermsunke>
24. How to use Taylor series to construct epsilon/delta limit proofs
for polynomials.
<http://forum.swarthmore.edu/epigone/sci.math/vexthooskel/gt7r3p...@forum.mathforum.com>
25. A proof that every derivative satisfies the intermediate value
property (IVP) and a stronger version of the IVP property that
every derivative satisfies.
<http://forum.swarthmore.edu/epigone/sci.math/zheldkheetwar/sz9nyp...@forum.mathforum.com>
26. Some remarks about pathological functions having the intermediate
value property are given in my comments to the original poster's
2'nd question.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/kayglanwhel/532qfp...@forum.swarthmore.edu>
27. Pathological strictly increasing continuous functions.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/yillundwax/av39qj...@forum.mathforum.com>
28. A list of web pages for nowhere differentiable continuous
functions.
<http://forum.swarthmore.edu/epigone/sci.math/jogruspo/7xl9rz...@forum.mathforum.com>
29. Historical discussion of Gilbert, Hankel, Schwarz, etc. during
the early 1870's as they grappled with the fact that a continuous
function can have a dense set of points of nondifferentiability.
<http://forum.swarthmore.edu/epigone/sci.math/phukheldpre/mgrudt...@forum.mathforum.com>
30. A proof that the maximal ideals in the ring of real-valued
continuous functions defined on the reals is the collection of
zero sets of singletons.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/zhahchumsmal/hwwb85...@forum.swarthmore.edu>
31. Remarks on obtaining a function from a given power series and
obtaining a power series from a given function. [See the
April 23, 2001 post by Dave L. Renfro.]
<http://forum.swarthmore.edu/epigone/ap-calc/kelddyshon>
32. Internet web pages dealing with x^x^x^...
<http://forum.swarthmore.edu/epigone/sci.math/veeclospor>
33. 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.
<http://forum.swarthmore.edu/epigone/alt.math.undergrad/shumtwabu/tihhui...@forum.swarthmore.edu>
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C. ADVANCED TOPICS [Roughly arranged this way: logic and set theory,
sequences and nets, metric and topological
properties of subsets of R^n, measure properties
of subsets of R^n, various real-valued function
properties, integrability issues, Baire category
and related results on spaces of functions and
spaces of sets.]
1. Some comments on almost disjoint sets.
<http://forum.swarthmore.edu/epigone/sci.math/spunblimpsming/1e5bix...@forum.swarthmore.edu>
2. A proof, using the fact that there exists a collection of c many
almost disjoint subsets of the positive integers (two proofs of
this result are given as well), that if V is an infinite
dimensional vector space over a field F, then the Hamel dimension
of the dual space V* is at least c. Several references are also
given for proofs that every infinite dimensional Banach space
has Hamel dimension at least c.
<http://forum.swarthmore.edu/epigone/sci.math/dwunshalclo/zecmf7...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/dwunshalclo/mn17qj...@forum.mathforum.com>
3. A proof that the set of real numbers is not a countable union of
sets each having cardinality less than 2^(aleph_0) using the fact
that every nonempty perfect set contains a pairwise disjoint
collection of 2^(aleph_0) many compact perfect sets.
<http://forum.swarthmore.edu/epigone/sci.math/clanfrendmel/oup56h...@forum.mathforum.com>
4. 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).
<http://forum.swarthmore.edu/epigone/sci.math/kahcrugror/zun9nh...@forum.swarthmore.edu>
5. Discussion of discontinuous solutions to f(x+y) = f(x) + f(y).
<http://forum.swarthmore.edu/epigone/k12.ed.math/swintendring/37b5af96....@news.gte.net>
6. 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". [Beardon's undergraduate level text
incorporates nets throughout.]
<http://forum.swarthmore.edu/epigone/sci.math/hilrulwa/2xsipv...@forum.mathforum.com>
7. 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.
<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal/f0ngz4...@forum.mathforum.com>
8. 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.
<http://forum.swarthmore.edu/epigone/sci.math/flarzhozal/8xk1ko...@forum.mathforum.com>
9. An historical essay on continuum hypothesis type results for
various classes of Borel and projective sets.
<http://forum.swarthmore.edu/epigone/sci.math/stantaxfan/u8ajjz...@forum.mathforum.com>
10. Some comments on Kakeya sets and the Kakeya Conjecture, some of
the more important published papers in the field, and some
references available on the internet.
<http://forum.swarthmore.edu/epigone/sci.math/blangchimsmen/64quin...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/blangchimsmen/wqq1fc...@forum.mathforum.com>
11. Some internet sites dealing with Apollonian packings.
<http://forum.swarthmore.edu/epigone/math-teach/zhugarsweld/m6cazr...@forum.swarthmore.edu>
12. Essay on the set of points of non-uniform convergence.
<http://forum.swarthmore.edu/epigone/sci.math/frorchestox/ua8ah1...@forum.mathforum.com>
13. Some historical remarks and references for Du Bois-Reymond's and
Cantor's method of condensation of singularities.
<http://forum.swarthmore.edu/epigone/sci.math/quinnoxlel/rxp8bg...@forum.swarthmore.edu>
14. Some comments on the origin of the phrase "almost everywhere".
<http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox/5pugbx...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox/qgurdh...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/historia_matematica/tinraxlox/p6cpv3...@forum.mathforum.com>
15. Discussion on using the Lebesgue density theorem to prove that a
set has measure zero (especially when the set is nowhere dense).
<http://forum.swarthmore.edu/epigone/sci.math.research/suntwatwi/i4gq5f...@forum.mathforum.com>
16. Some discussions on "points of non-measurability" of a set.
[I've since learned that this notion appeared in W. Alexandrow's
1915 Ph.D. Dissertation (see the last URL), which predates the
earliest references I give.]
<http://forum.swarthmore.edu/epigone/sci.math/nenvochoi/u5p6j0...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/nenvochoi/2rf8dj...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/nenvochoi/h9jpiy...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/historia_matematica/zhangjorprim/3978E8CA...@gateway.net>
<http://forum.swarthmore.edu/epigone/historia/verdtwonrou/96nwsm...@forum.mathforum.com>
17. Some biographical information about Evgenii P. Dolzhenko, the
person who introduced the notion of a sigma-porous set.
<http://forum.swarthmore.edu/epigone/historia_matematica/wopoosko/u33n09...@forum.mathforum.com>
18. Two definitions of "approximate limit", along with some
elementary remarks on approximate continuity and approximate
differentiation.
<http://forum.swarthmore.edu/epigone/sci.math/snugrepru/aogssc...@forum.mathforum.com>
19. References to pathological functions that satisfy the
intermediate value property: IVP functions that take every
value in every interval, IVP functions every value c-times in
every interval, IVP functions that take every value c-times in
every perfect set.
<http://forum.swarthmore.edu/epigone/sci.math/chimpderpin/6yqgsy...@forum.mathforum.com>
20. Example of a function satisfying the intermediate value property
that is not Riemann integrable over any closed subinterval of
[0,1]. [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/2000011817...@mimosa.csv.warwick.ac.uk>
21. Survey of some literature for "pathological curves", along with
some historical remarks.
<http://forum.swarthmore.edu/epigone/historia_matematica/shouzerdgleld/8cizd5...@forum.mathforum.com>
22. Some remarks about locally recurrent functions--nowhere constant
continuous functions all of whose level sets are perfect sets.
<http://forum.swarthmore.edu/epigone/sci.math/koxjalnil/snfdo2...@forum.mathforum.com>
23. Every monotone function is differentiable almost everywhere.
I give a detailed analysis of this measure zero exceptional
set using first category, porosity, Hausdorff dimension, and
Dini derivate notions.
<http://forum.swarthmore.edu/epigone/sci.math/queewirand>
24. A detailed survey of discontinuous derivatives.
<http://forum.swarthmore.edu/epigone/sci.math/pringquibroi>
25. A survey of results concerning the set of points at which a
measurable function has a symmetric derivative but not an
ordinary derivative.
<http://forum.swarthmore.edu/epigone/sci.math/bandglixkerm/1rylea...@forum.swarthmore.edu>
26. Some remarks about Riemann, Lebesgue, Henstock, and their
integrals.
<http://forum.swarthmore.edu/epigone/sci.math/gralfayfer/5gubr5...@forum.mathforum.com>
27. Remarks and references to the following result proved by
Sierpinski in 1910: If f^(x) is the oscillation of a function f
at x, then f^^ = f^^^ while f^ may differ from f^^.
<http://forum.swarthmore.edu/epigone/sci.math/bydwehkheh/9p9ip8...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/bydwehkheh/czomcw...@forum.mathforum.com>
28. A detailed survey of the fact that countable unions of closed
Lebesgue measure zero sets generate a sigma-ideal properly
contained in the sigma-ideal of sets simultaneously Lebesgue
measure zero and first category, along with the observation
that the exceptional set for the Riemann integrability
continuity condition is a countable union of closed Lebesgue
measure zero sets.
<http://forum.swarthmore.edu/epigone/sci.math/dwillydwul>
29. Baire category results for the set of normal numbers (mostly
literature references).
<http://forum.swarthmore.edu/epigone/sci.math/clargulgerm/9nd1k0...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math/clargulgerm/p5mqvh...@forum.swarthmore.edu>
30. Essay on the size of various subcollections of non-normal
numbers. [The set of normal numbers is large in the sense of
Lebesgue measure (its complement has measure zero), but small
in the sense of Baire category (it's a first category set).
Statements much stronger than "the normal numbers form a first
category set" are discussed.]
<http://forum.swarthmore.edu/epigone/sci.math/yeplaxdum/getei6...@forum.mathforum.com>
31. Baire category results involving divergent rearrangements of
a conditionally convergent series. ["Most" rearrangements of
each conditionally convergent series of real numbers have
every extended real number as a cluster point (i.e. "most"
rearrangements of any conditionally convergent series diverge
in the worst way).]
<http://forum.swarthmore.edu/epigone/sci.math/snysmermbror/1cus2w...@forum.mathforum.com>
32. In the space of continuous functions with the sup norm, some
remarks are made about the density of the set of nowhere
differentiable continuous functions and about the density of
the set of functions whose graphs have a specified Hausdorff
dimension.
<http://forum.swarthmore.edu/epigone/sci.math.research/smarclurerd/n4yb1r...@forum.swarthmore.edu>
33. A list of references on Haar null sets and a summary of some
results in Hongjian Shi's 1997 Ph.D. Dissertation (under Brian
Thomson) "Measure-Theoretic Notions of Prevalence".
<http://forum.swarthmore.edu/epigone/sci.math.research/cleekhenbrum/9oz8hl...@forum.swarthmore.edu>
<http://forum.swarthmore.edu/epigone/sci.math.research/cleekhenbrum/18cc8u...@forum.swarthmore.edu>
34. A discussion (with lengthy list of references) for some of the
various applications of fixed point theorems.
<http://forum.swarthmore.edu/epigone/sci.math/dawhimge/nzam9s...@forum.mathforum.com>
<http://forum.swarthmore.edu/epigone/sci.math/dawhimge/fgzmso...@forum.mathforum.com>
35. A discussion about the duality of two types of mathematical
constructions. One is a combinatorial "from below" construction
of subgroups, subfields, etc. The other is an impredicative
"from above" construction.
<http://forum.swarthmore.edu/epigone/math-teach/fraxvimpcrim>
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