# Math 402 – Modern Analysis, Winter 2019

Instructor: Anatolii Grinshpan
Office hours:  MW 12-1 and R 2-3, Math Resource Center.

Jan 7.  Introduction to the course. Darboux’s property. Rolle’s theorem.

Jan 9.  Mean-value theorems.

Jan 11. Bernoulli - l’Hôpital rules.

Reading: 5.2, 5.3. Homework 1.

Jan 14. Continuous nondifferentiable functions.

Jan 16. Blancmange function: construction and properties. Notes.

Jan 18. Functional sequences: pointwise convergence.

Reading: 5.4, 6.2. Homework 2.

Jan 21. MLK day (no class).

Jan 23. No class.

Jan 25. Functional sequences: uniform convergence. Cauchy criterion. Example.

Reading: 6.2. Homework 3.

Jan 28. Quiz 1 (pointwise and uniform convergence). Stokes-Seidel theorem.

Jan 30. Uniform convergence and differentiation. Notes.

Feb  1. Series of functions. Example.

Reading: 6.3, 6.4. Homework 4.

Feb  4. Quiz 2 (series of functions). Uniform convergence and integration.

Feb  6. Real power series: absolute and uniform convergence.

Feb  8. Midterm 1 (5.3, 5.4, 6.2-6.4).

Reading: 6.5. Homework 5.

Feb 11. Abel’s theorem. Midterm discussion.

Feb 13. Abel’s test for uniform convergence. Notes.

Feb 15. Taylor polynomials. Peano and Lagrange remainder terms. Notes.

Reading: 6.5, 6.6. Homework 6.

Feb 18. Quiz 3 (power series). Convergence of Taylor series.

Feb 20. Snow.

Feb 22. Weierstrass approximation and Bernstein polynomials. Notes.

Reading: 6.6-6.8. Homework 7.

Feb 25. Properties of Darboux sums.

Feb 27. Darboux sums and Riemann-integrability. Notes.

Mar  1. Quiz 4 (Darboux sums). Outer content of a set.

Reading: 7.1-7.3. Homework 8.

Mar  4. Midterm 2 (6.5-6.7, 7.1-7.3).

Mar  6. Properties of Riemann-integrable functions. Notes.

Mar  8. The Fundamental Theorem of Calculus.

Mar 11. Quiz 5 (Riemann integral). Set of discontinuities. Notes.

Mar 13. Lebesgue null sets. Property of α-continuity. Worksheet.

Mar 15. Lebesgue’s criterion for Riemann-integrability. Notes.