Introduction to Analysis (Math420)

Content

Textbook: Jiri Lebl: Basic Analysis. Custom Pitt edition.

Prerequisites

The course concerns the One-Variable Calculus. Prerequisite is Math 413: Introduction to Theoretical Mathematics. If you do not feel comfortable with the prerequisite material, please contact the instructor in the beginning of the course.

Homework

Homework will be assigned each Monday (starting in the second week of the semester), and it will be due on Tuesday the following week during the recitations. Late homework will not be accepted. The solution of each exercise will be evaluated in the scale 0-5 points, taking into account the correctness, clarity and neatness of presentation.

Grades

Your final grade depends on your performance on the final exam as well as on your total grade. Grades will be based on homework (35%), two midterms (15% + 15%) and the final (35%). There will be no make up midterm exams. If you miss the midterm exam for a documented medical reason, your grade on it will be the prorated grade of your final exam. Incompletes will almost never be given, and only for cases of extreme personal tragedy.

Core topics

  1. The Bolzano-Weierstrass Theorem; Cauchy sequences; Cauchy completeness of the real numbers.
  2. Real-valued functions on an interval: limits and continuity.
  3. Intermediate Value Theorem; Max-Min Theorem.
  4. Uniform continuity; continuous functions on a closed and bounded interval are uniformly continuous.
  5. Differentiable functions.
  6. Interior Extremum Theorem, Rolle’s Theorem, Mean Value Theorem.
  7. Taylor’s Theorem and Taylor Series.
  8. The Riemann Integral on a closed and bounded interval.
  9. The Fundamental Theorem of Calculus.
  10. Definition and examples of pointwise and uniformly convergent sequences of functions.
  11. Continuity of uniform limits of continuous functions.
  12. Interchange of uniform limits and integration.
  13. Interchange of limits with differentiation.
  14. The M-test for uniform convergence of series.
  15. Application to power series.

Class hours

tba

Office hours

Wednesday tbd Thackeray Hall, Room 410

Further office hours available upon request by email: armin@pitt.edu

Teaching Assistant

Teaching Assistant. tba (recitations tba)

Exercise Sheets

  • tba (due: Oct tba)