COURSE NUMBER AND TITLE: MATH 4011 Real Variables I

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of the real number system and functions. Topics include sequences, limits, continuity, differentiation and integration.

PREREQUISITE(S): MATH 3280

SUGGESTED TEXT(S): A First Course in Real Analysis, by Protter and Morrey

COURSE OUTLINE:

1. The Real Number System
   • Axioms for a Field
   • Natural Numbers and Sequences
   • Mathematical Induction
2. Continuity and Limits
   • Continuity
   • Limits
3. Basic Properties of Functions on R1
   • The Intermediate-Value Theorem
   • Least Upper Bounds; Greatest Lower Bounds
   • The Bolzano-Weierstrass Theorem
   • The Boundedness and Extreme-Value Theorems
   • Uniform Continuity
   • The Cauchy Criterion
4. Elementary Theory of Differentiation
   • The Derivative in R1
   • Inverse Functions in R1
5. Elementary Theory of Integration
   • The Darboux Integral for Functions on R1
   • The Riemann Integral
   • The Fundamental Theorem of Calculus
6. Elementary Theory of Metric Spaces
   • The Schwarz and Triangle Inequalities; Metric Spaces
   • Elements of Point Set Topology
   • Countable and Uncountable Sets
   • Compact Sets and the Heine-Borel Theorem