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.

Bartle and Sherbert. The course might cover the

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