MAT378, Foundations of Analysis

Winter 2010

Lectures: Tuesdays 1-3 and Thursdays 2-3 at NE174.

Instructor: Professor Ilia Binder (, South Building 4038, Phone: (905) 569-4381.

Office Hours: Tuesdays, 3-4; Thursdays, 1-2; and by appointment.

New"Prefinal" Office Hours: Monday, April 19, 1-3pm.

TA: Mario Morfin (

Tutorial: Fridays, 1-2 at SE 1142.

Required Text:  Real analysis with real applications, by Davidson and Donsig.
The book has been re-printed by University of Toronto Press.

Course notes:
Introduction: preview of coming attractions.
Chapter 1: Review.
Chapter 2: Limits and Real numbers.
Chapter 3: Metric spaces.
Chapter 4: Differentiation and Integration.

Appendix: Limits of functions.

Course website:

Prerequisites: MAT102H5, (223H5, 224H5)/248Y5, 212H5/242H5/258Y5.

Topics. The course is the rigorous introduction to Real Analysis. We start with the careful discussion of real numbers, the basic concepts of limits, continuity, Riemann integrability, and differentiability. More advanced topics will include the foundations of topology, the study of the normed and metric spaces, Fourier series, and the foundations of the Measure theory.

Homework. Weekly homework assignments will be due on Thursdays at the beginning of the class, starting January 14. The assignments will be posted the preceding Thursdays on the course website.

Assignment #1, due January 14.

Assignment #2, due January 21: Section 2.2, Problem C; Section 2.4, Problems C, K; Section 2.5, Problems C, E.

Assignment #3, due January 28: Section 2.3, Problems A, D; Section 2.6, Problem G; Section 2.8, Problems C, I.

Assignment #4, due February 4: Section 2.7, Problems D, I; Section 3.1, Problem C; Section 3.2, Problems I, Q.

Assignment #5, due February 11: Section 4.1, Problem C; Section 4.2, Problem C; Section 4.3, Problem N (do it for an arbitrary metric space), Section 9.1, Problems B, E.

Assignment #6, due March 4: Section 4.3, Problem L; Section 4.4, Problem F (do it for an arbitrary compact set K); Section 5.1, Problem E; Section 5.3, Problem H; Section 9.2, Problem A.

Assignment #7, due March 11: Section 5.4, Problems G, J; Section 5.5, Problems C,I; Section 9.2, Problem J.

Assignment #8, due March 18: Section 5.6, Problem F; Section 6.1, Problem S; Section 6.2, Problem I; Section 6.3, problem M; Section 9.4, problem F.

Assignment #9, due April 1: Section 6.3, Problem Q (you can use Lebesgue Theorem here); Section 6.4, Problems G, J; Section 6.6, Problems A, F.

Practice problems for Midterm #1(do not turn in): 1.2.F, 1.6.A, 2.2.I, 2.3.C, 2.4.J, 2.5.F, 2.6.H, 2.7.H, 2.8.F, 3.1.F, 3.2.G, 4.1.F, 4.2.F, 4.3.G, 9.1.A.

Practice problems for Midterm #2(do not turn in): 4.3.J (for an arbitrary metric space), 4.4.G, 5.1.G, 5.3.L, 5.4.H, 5.5.H, 5.6.D, 6.1.E, 6.2.H.a),6.3.G, 9.2.F, 9.4.E.

Additional practice problems for the Final(do not turn in): 6.4.I, 6.6.I, 8.1.H, 8.2.D, 8.3.C.

Midterm Tests. There will be two in-class midterm tests: on Thursday, February 11 and on Thursday, March 18. Textbooks, notes, or calculators are not allowed for these tests.

Final exam. The final exam will be held on Tuesday, April 20, 12-3pm, at SE Cafe C. You will be allowed to use one one-sided letter-sized page of notes. Textbooks or calculators are not allowed for this exam.

Grading. Grades will be based on regular weekly homework (15%), midterm tests (20% each) and the final  exam(45%). The two lowest homework grades will be dropped. I will also occasionally assign bonus problems. Late homework will not be accepted. There will be no make-up midterm tests or final exam, and an undocumented absence will result in zero credit.

Plagiarism and Academic Honesty. Students are expected to adhere to the academic regulations of the University as outlined in the “Code of Behaviour on Academic Matters” which can be found in the UTM Calendar or on the web at

The work you submit must be your own and cannot contain anyone else’s work or ideas without proper attribution. Plagiarism is a form of academic fraud and is treated very seriously. Please have a look at