UNIVERSITY OF MADRAS
B.Sc. DEGREE COURSE IN MATHEMATICS
SEMESTER SYSTEM WITH CREDITS
(Effective from the Academic Year 2003-2004)

SYLLABUS

Semester VI - Paper XIV - Real Analysis II

Duration of Examination: 3 hrs
Maximum Marks: 100
Credits: 4

Continuous functions on Metric Spaces

Functions continuous at a point on the real line,reformulation,functions continuous on a metric space, open sets, closed sets, discontinuous functions on the real line.

Connectedness completeness and compactness

More about open sets, connected sets,bounded sets and totally bounded sets, complete metric spaces, compact metric spaces, continuous functions on a compact metric space, continuity of inverse functions, uniform continuity.

Calculus

sets of measure zero, definition of the Riemann integral, existence of the Riemann Integral(statement only) properties of Riemann integral, derivatives, Rolle's theorem,. Law of mean, Fundamental theorems of calculus, Taylor's theorem. .

Sequences and Series of Functions.

Pointwise convergence of sequences of functions, uniform convergence of sequences of functions.

Treatment as in "Methods of Real Analysis"-Richard R.Goldberg.Oxford and IBH publishing Co.

chapter 5 and chapter 6 full, chapter 7 (section 7.1-7. 8), chapter 8 (section 8.5 only), chapter 9(sections 9.1 and 9.2 only). R. Bartle and Sherbert. Real Analysis fifth edition, Wiley and sons, New York.