I. Number and Title of Course

MAT 417 - Introduction to Real Analysis I

We have updated the bibliography to better reflect the current resources used in the course, and we have redefined the major objectives so that they are now in terms of student outcomes. The course continues to serve the following purposes in our program:

A. To provide students with opportunities to learn proofs of the theorems of calculus in a formal, rigorous way.

B. To provide students who plan to teach mathematics in secondary schools with a sound background in mathematical analysis so that they may be qualified to teach calculus.

C. To provide students with opportunities to learn proofs of the theorems of calculus in a formal, rigorous way.

A. Students will gain an understanding of the foundations of the real number system and its axioms.

B. Students will gain familiarity with proofs of theorems? surrounding limits, functions, continuity, sequences, series, differentiation, and the Riemann integral.

C. Students will develop skills in constructing proofs.

A. Preliminaries

1. Axioms of order

2. Well-ordering principle

3. Axiom of completeness (or completeness property of R)

4. First and second principle of Mathematical Induction

5. Countable and uncountable sets

B. Functions, Limits, Sequences

1. Relations and Functions

2. Limit of a sequence

3. Basic limit theorems for sequences

4. Cauchy sequence, bounded sequence

5. Limits of functions

6. Limit theorems for functions

C. Continuity

1. Definition

2. Basic theorems on continuity

3. Types of discontinuity

4. Monotonic functions and their inverses

5. More theorems for continuous functions (on closed intervals)

6. Uniform continuity

D. Differentiation

1. Definition of the Derivative

2. One-sided derivatives

3. Rolle's Theorem, Mean Value Theorem and applications

4. The Extended Mean Value Theorem and its application

5. Differentials and approximation by differentials

6. L 'Hospital 's Rule

E. Riemann Integral

1. The Riemann Integral

2. The Fundamental Theorem of Calculus

3. Integration by substitution

4. Improper integrals (on finite and infinite intervals)

5. Comparison tests

F. Infinite series, power series

1. Basic definitions

2. Basic theorems

3. Convergence tests (integral tests, comparison test, ratio and root test)

4. Alternating series

5. Absolute and conditional convergence

6. Rearrangement

7. Interval of convergence

8. Taylor series and Taylor's Theorem

9. Integration and differentiation of power series

Bartle, G.& Sherbert, R., Introduction to Real Analysis, John Wiley & Sons, New York, NY, 1990.

Bilodean, G. & Thie, R., An Introduction to Analysis, McGraw-Hill, New York, NY, 1995.

Bruckner, A. et al., Real Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1997.

Caplan, Wilfred, Advanced Calculus, Addison-Wesley, Reading, MA, 1984.

Fitzpatrick, P., Advanced Calculus: A Course in Mathematical Analysis, Brooks/ Cole, Belmont, CA, 1996.

Fulks, D. Advanced Calculus, 4th Ed., John Wiley & Sons, New York, NY, 1988.

Gariepy, R. & Ziemer, W., Modern Real Analysis, Brooks/ Cole, Belmont, CA, 1995.

Gaughan, E., Introduction to Analysis, 5th Ed., Brooks/ Cole, Belmont, CA, 1998.

Gordon, R., Real Analysis: A First Course, Addison Wesley, Reading, MA, 1997.

Kaplan, W., Advanced Calculus, 4th Ed., Addison Wesley, Reading, MA, 1992.

Kirkwood, J., An Introduction to Analysis, 2nd Ed., Brooks/ Cole, Belmont, CA, 1995.

Lay, S., Analysis with an Introduction to Proof, Prentice-Hall, Englewood Cliffs, NJ, 1990.

Royden, H.L., Real Analysis, 3rd Ed., Macmillan, New York, NY, 1988.

Stoll, M., Introduction to Real Analysis, Addison Wesley, Reading, MA, 1997.

A. Presentation by lecture and discussion.

B. Evaluation by in-class examinations.

MAT 263 (MAT 270 is recommended)

3 credits: (3:0)

This course proposal was examined in accordance with recommended procedures and was approved by the Department of Mathematics.

_______________________________ Signature of Department Chair

A rigorous treatment of elementary real analysis, including: properties and axioms of the real number system, relations and functions, sequences, continuity, differentiation, infinite series, power series, and the Reimann integral.

A. An M.A. or M.S. degree in Mathematics, at least 6 graduate credit hours of Real Analysis, and Graduate Faculty status are required to teach this course.

B. Some of the faculty who may teach this course are:

Peter Mercer, Ph.D. in Mathematics

Daniel Cunningham, Ph.D. in Mathematics

Joaquin Carbonara, Ph.D. in Mathematics

Present classroom facilities and resources available in Butler Library are adequate.