Prerequisites & Bulletin Description
Course Objectives
The principal aim of Real Analysis I is for students to learn how to carry out a rigorous analysis of calculus of functions of a real variable. Students are expected to learn how to write, in a logical manner, important theorems and properties of continuous, differentiable and integrable functions. Students learn to solve problems using the concepts of analysis. They present their solutions as rigorous proofs written in correct mathematical English. Students will be able to devise, organize and present brief (half-page) solutions based on definitions and theorems of analysis. Students who successfully complete this course will be capable of applying the Completeness Axiom. They will also be able to prove and apply the Bolzano-Weierstrass Theorem, the Intermediate Value Theorem, the Mean Value Theorem, the Inverse Function Theorem and the Fundamental Theorem of Calculus. They will be able to define uniform continuity and be able to prove that continuous functions defined on closed bounded intervals are uniformly continuous. Students will also demonstrate that rational, trigonometric, exponential and logarithmic functions are differentiable. They will be able to construct differentiable functions that are not continuously differentiable. Students will be able to determine if a function has a continuous or differentiable inverse. Students will be able to prove that continuous functions are Riemann integrable.
Evaluation of Students
Students will be graded on their ability to devise, organize and present complete solutions in correct mathematical English. While instructors may design their own methods of evaluating student performance, these methods must include in-class examinations, graded homework assignments and a final exam.
Course Outline
| Topics | Number of Weeks |
|---|---|
| The Real Number System (R): ordered field axioms, the Well-Ordering Principle, the Completeness Axiom, countability. | 2 |
| Sequences in R: limit theorems, Bolzano-Weiertrass Theorem, Cauchy sequences. | 3 |
| Continuity on R: two and one-sided limits of functions, limits of functions at infinity, continuity, uniform continuity. | 2 |
| Differentiability on R: the derivative, differentiability theorems, the Mean Value Theorem, L'Hospital's Rule, monotone functions, the Inverse Function Theorem. | 4 |
| The Riemann integral: Riemann sums, The Fundamental Theorem of Calculus. | 4 |
Textbooks & Software
Abbott, S., Understanding Analysis, 2001
Lay, S., Analysis with Introduction to Proof, 3rd Edition, Pearson Education, 2000
Marsden, J., Hoffmann M., Elementary Classical Analysis, 2nd Edition, W.H. Freeman and Co., 1993
Wade, W., An Introduction to Analysis, 3rd Edition, Prentice Hall, 2003
Submitted by: David Ellis and Eric Hayashi
Date: June 21, 2004