Mathematical Analysis

Basic information

Workload: 

45 hours

Syllabus: 

Enumerability; concept of infimum and supremum; construction of real numbers. Sequences and numerical series: notion of limit, Monotonous Sequence Theorem, Embedded Intervals Theorem, Cauchy sequence, Bolzano-Weierstrass Theorem, convergence criteria. Line topology: open, closed, convex, connected sets. Real functions. Limit. Continuity: Bolzano's Theorem, Intermediate Values Theorem, Weierstrass Theorem, Uniform continuity. Derivation: monotony, Mean-Value Theorem, L'Hôpital Rules, Taylor, optimality criteria. Riemann integral; Fundamental Theorem of Calculus; Mean-Value Theorem for Integrals. Mention to the integral and Lebesgue measure. Sequences and series of functions: punctual, uniform convergence, limit under the integral sign. 

Bibliography

Mandatory: 

•    Lima, E. L. Analysis Course. Volume 1. Projeto Euclides, IMPA, 2000. 
•    Rudin, Walter. Principles of mathematical analysis. Vol. 3. New York: McGraw-hill, 1964. 
•    Neri, Cassio and Cabral, Marcos; Real Analysis Course; IM-UFRJ, 2011. 

Complementary: 

•    Thomson, B.S., Bruckner, J.B. and Bruckner, A.M. Elementary real analysis. 2008. ClassicalRealAnalysis.com 
•    Lages, Elon. Real analysis. Volume 1. University Mathematical Collection, IMPA, 1989. 
•    Abbott, Stephen. Understanding analysis. New York: Springer, 2001. 
•    Apostol, T. M. Mathematical analysis. Addison-Wesley Reading, 1964. 
•    Rudin, W., Real and Complex Analysis, McGraw-Hill Book Company Inc., New York-Toronto-London, 1974.