Introductn to the Theory of Functions I
MATH 765 and MATH 766 are theoretical courses on the fundamental concepts of analysis and the methods of proof. These two courses include the concept of a real number; limits, continuity, and uniform convergence; derivatives and integrals of functions of one and of several real variables.
Intro to Analysis, Wade, Prentice Hall, 4th edition.
MATH 223 and MATH 290 or equivalent.
- The Real Number System
- Field properties, well-ordering and completeness
- Sequences and Bolzano-Weierstrass theorem
- Functions and countability
- Continuity and Differentiability on the Real Line
- Limits, continuity and uniform continuity
- Differentiability, mean value theorem and monotonicity
- The Riemann Integral
- The Riemann integral and Riemann sums
- The fundamental theorem of calculus, integration by parts and change of variables
- Improper Riemann Integrals
- Infinite Series
- Convergence, absolute convergence and tests of convergence
(Torres 2017 )