Real Analysis

SYLLABUS- REAL ANALYSIS,
Unit-1
Axiomatic study of real numbers, Completeness property in R, Archimedean
property, Countable and uncountable sets, Neighbourhood, Interior points, Limit
points, Open and closed sets, Derived sets, Dense sets, Perfect sets, Bolzano-
Weierstrass theorem.
Unit-2
Sequences of real numbers, Subsequences, Bounded and monotonic sequences,
Convergent sequences, Cauchy's theorems on limit, Cauchy sequence, Cauchy's
general principle of convergence, Uniform convergence of sequences and series of
functions, Weierstrass M-test, Abel's and Dirichlet's tests.
Unit-3
Sequential continuity, Boundeness and intermediate value properties of continuous
functions, Uniform continuity, Meaning of sign of derivative, Darboux theorem.
Limit and continuity of functions of two variables, Taylor's theorem for functions of
two variables, Maxima and minima of functions of three variables, Lagrange's method
of undetermined multipliers.
Unit-4
Riemann integral, Integrability of continuous and monotonic functions,
Fundamental theorem of integral calculus, Mean value theorems of integral calculus,
Improper integrals and their convergence, Comparison test, m-test, Abel's test,
Dirichlet's test, Integral as a function of a parameter and its differentiability and
integrability.
Unit-5
Definition and examples of metric spaces, Neighbourhoods, Interior points, Limit
points, Open and closed sets, Subspaces, Convergent and Cauchy sequences,
Completeness, Cantor's intersection theorem.