Department of Mathematics

Patricia Bauman

Math 440 Fall 2017

Professor: Patricia Bauman

Office: Math 718
Phone: 494-1945
Office Hours: Monday and Wednesday, 1:30-2:30 pm, or by appointment.

Course Lectures:
MWF 9:30-10:20 in REC 317.

Textbook: The Elements of Real Analysis, Second Edition, by Robert G. Bartle, John Wiley and Sons, 1975.

Syllabus: Most of Chapters I-V in the text.

Recommended Extra Material: Appendix A of the book by Bartle and Sherbert, 4th edition, on Logic and Proofs. (Note that this is not the same book as our textbook.)

Homework will be collected weekly in class on Wednesdays (with some exceptions). No late homeworks will be accepted; however, the lowest homework score will be dropped. The assignments are posted below.

Exams: There will be two midterm exams (in class) and a comprehensive final exam.
-- Exam 1 is on Wednesday, Oct. 4, in class. Open textbook (with no notes in the textbook). Covers Sec. 3-12.
-- Exam 2 is on Wednesday, Nov. 15, in class. Open textbook (with no notes in the textbook). Covers Sec. 14-17, 20, 22.

Course Grade: Your course grade will be computed by the scheme:
Final Score = 22% Homework + 23% Exam 1 + 23% Exam 2 + 32% Final Exam.

Topics Covered:
Mon, Aug. 21: Section 4 Algebraic Properties of R.
Wed, Aug. 23: Section 5 Order Properties of R.
Fri, Aug. 25: Section 6 The Completeness Property of R.
Mon, Aug. 28: Finished Section 6.
Wed, Aug. 30: Section 3 Finite and Infinite Sets.
Fri, Sept. 1: Section 7 Intervals and the Cantor Set. Construction of R as Dedekind Cuts. Nested cells property of R.
Mon, Sept. 4: No class (Labor Day).
Wed, Sept. 6: Section 8 Vector Spaces, Cartesion Spaces, Inner Products, Norms.
Fri, Sept. 8: Finished Sec 8. Section 9: Open and Closed Sets in R^p.
Mon, Sept. 11: Neighborhoods, Interior, Exterior and Boundary points in R^p.
Wed, Sept. 13: Finished Sec 9. Section 10: Nested Cells.
Fri, Sept. 15: Cluster points, Isolated Points, Nested Cells Property of R^p. Bolzano-Weierstrass Theorem.
Mon, Sept. 18: Finished Sec 10. Section 11: Compact Sets in R^p.
Wed, Sept. 20: Direct Proofs of Compactness. Heine-Borel Theorem.
Fri, Sept. 22: Applications of the Heine-Borel Theorem. Finished Sec 11. Started Section 12: Connected Sets in R^p.
Mon, Sept. 25: Connected sets in R and R^p.
Wed, Sept. 27: Finished Sec 12. Started Sec. 14: Intro to Sequences in R^p.
Fri, Sept. 29: Theorems on Convergence of Sequences in R^p.
Mon, Oct. 2: Examples on Convergence of Sequences in R^p. Finished Sec. 14. Started Sec. 15: Subsequences and Combinations.
Wed, Oct. 4: Exam 1 on Sections 3-12.
Fri, Oct. 6: Finished Sec. 15.
Mon, Oct. 9: No class- Oct. Break.
Wed, Oct. 11: Section 16: Two Criteria for Convergence of Sequences: Monotone Convergence Theorem in R. Application: The Definition of e.
Fri, Oct. 13: Discussion of Exam 1.
Mon, Oct. 16: Bolzano-Weierstrass Theorem for Sequences in R^p. Cauchy Sequences in R^p.
Wed, Oct. 18: Cauchy Convergence Criterion Sequences in R^p. Finished Sec 16. Started Section 17: Sequences of Functions on Domains in R^p to R^q, Pointwise Convergence and Examples.
Fri, Oct. 20: Uniform Convergence of Sequences of Functions on Domains in R^p to R^q, Necessary and Sufficient Conditions for when a Sequence of Functions does not Converge Uniformly, Examples.
Mon, Oct. 23: The sup norm (uniform norm) for Bounded Functions on a Domain in R^p to R^q. Finished Sec 17. Started Section 20: Local Properties of Continuous Functions from D(f) in R^p to R^q.
Wed, Oct. 25: Equivalent Definitions of Continuity at a point. Discontinuity Criterion.
Fri, Oct. 27: Examples of Continuous and Discontinuous Functions at a Point, Epsilon-delta Proofs. Continuity of Combinations of Continuous Functions at a Point, including Polynomials and Rational Functions on R^p.
Mon, Oct. 30: Continuity of Compositions of Continuous Functions. Finished Sec. 20. Started Section 22: Global Continuity Theorem.
Wed, Nov. 1: Preservation of Connectedness: The Continuous Image of a Connected Set in R^p is Connected.
Fri, Nov. 3: Bolzano's Intermediate Value Theorem, Preservation of Compactness, Maximum and Minimum Value Theorem.
Mon, Nov. 6: Continuity of the Inverse Function. Finished Sec. 22. Started Section 23: Uniformly Continuous Functions in R^p, Characterization of Functions that are not Uniformly Continuous, Examples.
Wed, Nov. 8: Uniform Continuity Theorem, Continuity of Uniform Limits of Continuous Functions in R^p. Finished Sec 23. Section 24-first part: Continuity of the Uniform Limit of a Sequence of Continuous Functions in R^p.
Fri, Nov. 10: Section 25: Limit (Deleted Limit) of a Function at a Cluster Point of Its Domain. Equivalent Definitions. Limits of Combinations of Functions (Sums, Products, etc.). Continuity (at a Cluster Point of the Domain) in terms of Limit.
Mon, Nov. 13: Section 27: Rolle's Theorem, Mean Value Theorem, Interior Maximum/Minimum Theorem.
Wed, Nov. 15: Exam 2 on Sections 14-17, 20, 22.
Fri, Nov. 17: Sec. 27-28: Applications of the Mean Value Theorem. Cauchy Mean Value Theorem. Interchange of Limit and Derivative.
Mon, Nov. 20: Taylor's Theorem. Started Section 29: Riemann-Stieltjes Integral.

Wed, Nov. 22: No class (Thanksgiving Break)
Fri, Nov. 24: No class (Thanksgiving Break)

Homework Assignments:
1.) Due Wed, Aug. 30: 4F,4H,5C,5E,5F,6D,6G,6H.
2.) Due Fri, Sept. 8: 3D,3G,7F,7G, 7I (first statement only), 7J.
3.) Due Wed, Sept. 13: 8F,8G,8M,8Q.
4.) Due Wed, Sept. 20: 9C,9G,9L,9Q(a), 10C, 10F.
5.) Due Wed, Sept. 27: 9M, 11A,11C,11J,11K, 12A,12C. (Hint: You can use 9M to do 12C)
6.) Due Wed, Oct. 11: 14D,14G,14H,14O, 15C(b,f).
7.) Due Wed, Oct. 18: 15F,15M,15N, 16A.
8.) Due Wed, Oct. 25: 16I,16J,16P,16M(b), 17D,17L,17M.
9.) Due Wed, Nov. 1: 17P,17Q, 20A,20E,20F,20J, 22A.
10.) Due Wed, Nov. 8: 22C,22D,22E,22K,22N,22O.
11.) Due Fri, Nov. 17 23E,23F(a,c), 24A,24C,24E(a,c,d).
12.) Due Wed, Nov. 29: 25A(a,e),25I,25J, 27A(d),27C,27D, 28E,28K.

Home || Courses || Preprints || Publications