M361K Syllabus

INTRODUCTION TO REAL ANALYSIS

Prerequisite and degree relevance: Either consent of Mathematics Advisor, or two of M341, 328K, 325K (Philosophy 313K may be substituted for M325K), with a grade of at least C-. May not be counted by students with credit for M365K with a grade of C or better.

Course description: This is a rigorous treatment of the real number system, of real sequences, and of limits, continuity, derivatives, and integrals of real-valued functions of one real variable.

Text: A reasonable text is Introduction to Real Analysis by Bartle and Sherbert. The course might cover the bulk of chapters one through six in that book.

Topics:

  • The real number system: the axiomatic description of the real number system as the unique complete ordered field, with special emphasis on the completeness axiom; the elementary topology of the real line.
  • Real sequences: the definition and elementary properties of sequential limits; subsequences and accumulation points; monotone sequences; inferior and superior limits; the Bolzano-Weierstrass theorem.
  • Limits and continuity of functions: the definition and elementary properties of limits of functions, including the usual variations on the basic theme (e.g.,one-sided limits, infinite limits, limits at infinity); continuity; the funtdamental facts concerning continuous functions on intervals (e.g., Intexmexliatc Value Theorem, Maximum-Minimum Theorem, continuity of inverse functions, uniform continuity on closed intervals).
  • Differentiation: the definition and geometric significance of the derivative; differentiation rules; the Mean Value Theorem and its consequences; Taylor's Theorem; L'Hospital's rules; convexity.
  • Riemann Integration: the definition and elementary properties of the Ricmann integral; the integrability of continuous functions and monotone functions; the Fundamental Theorems of Calculus.

March, 1989