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


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


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


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


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


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


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


       8. Joel Feinstein's handouts and lecture notes in (a) real analysis
          and (b) metric and topological spaces [Univ. of Nottingham]


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


       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]


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


       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.


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


       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.)


       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)]


       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 inequal! ity"]


       18. Michael Filaseta's notes for Math 555: Real Analysis II [Univ.
           of South Carolina]


       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".]


       20. Math 105A: Real Analysis [Univ. of California at Santa Cruz]


       21. Yuri Safarov's notes for Real Analysis CM 321A [King's College


       22. Karen E. Donnelly's notes for M345 Real Analysis [Fall 2000 at
           Saint Joseph's College]


       23. Erhan Cinlar and Robert J. Vanderbei's book "Real Analysis for
           Engineers" (483 K .pdf file, 119 page output)


       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)


       25. Notes for MATH 2610 HIGHER REAL ANALYSIS - Session 1, 2001
           at Univ. of New South Wales


       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)]


       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)


       28. Jim Langley's notes for real analysis [Univ. of Nottingham]
           (393 K .pdf file)


       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")


       30. Leif Abrahamsson's course material for real analysis
           [Uppsala Univ., Sweden]




       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.


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


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


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


       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.]


          listed in Section V: "papers, projects, essays by or for students"
          involve real analysis topics.]


       7. Chris Hillman and Roland Gunesch's resources for Entropy in
          Ergodic Theory and Dynamical Systems


       8. Terence Tao's "Harmonic Analysis Links"


       9. Online Books and Lecture Notes in Mathematics


       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.]


       11. Michael Botsko's [Saint Vincent College] "A Web Page in Real





       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.


       2. 9 applications of rationalizing the numerator or denominator.
          [See the June 13, 2001 post by Dave L. Renfro.]


       3. Why we don't prove a trig. identity as if we were solving an
          equation. [Correction given in 2'nd URL.]


       4. Why don't we use negative bases for exponential functions?


       5. Remarks about Carl Sagan's novel "Contact", messages in Pi, and
          normal numbers.


       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.]


       7. Some examples where local max/min's of [f(x)]^[g(x)] can be
          explicitly found.


       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.]


       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.]


       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.]


       11. Complete details for the integration of (1 + x - x^2)^(1/2).


       12. How to integrate (1 + x^4)^(-1). [Correction given in 2'nd URL.]


       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.


       14. How to evaluate integrals from -infinity to +infinity of
           (polynomial)*exp(-a*x^2 + b*x + c) for a > 0.


       15. Comments and references about series that converge or diverge
           very slowly.


       16. The Taylor series method for evaluating limits.


       17. A survey of a lot of elementary ways to evaluate limits.


       18. Concerning the limit [tan(sin x) - sin(tan x)]/(x^7) as x --> 0.


       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.


       20. How to prove 1=2 by partial differentiation.



       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, inte! grability issues.]

       1. Three proofs that a set with n elements has 2^n subsets.


       2. Some web pages having examples of mathematical induction proofs.


       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.]



       4. A list of 16 web pages dealing with elementary aspects of the
          cardinality of sets, along with my comments on them.


       5. Some texts on elementary set theory and my comments on them.


       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.


       7. Some references for Cohen's CH independence result for someone
          just beginning (plus some related internet sites).


       8. Proofs that there are infinitely many primes using relatively
          advanced mathematical ideas.


       9. References for the prime number theorem and the divergence of
          the harmonic prime series.


       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.


       11. Proof that a trig. function of a rational number of degrees is
           an algebraic number, along with several literature references.


       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.


       13. Remarks about explicit algebraic numbers and algebraic numbers.

       <http://forum.swarthmore.edu/epigone/sci.math.symbolic/playzerdblar/k! f2fw9x8n7f7@forum.mathforum.com>

       14. Two proofs that any closed set of real numbers is equal to the
           set of cluster points of some sequence.


       15. Some fractal dimension, and other, results concerning
           Cantor sets constructed by a dyadic process.


       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.


       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.


       18. A discussion of various notions of "increasing at a point" vs.
           "increasing on an interval" vs. "having a positive derivative".


       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.]