MASTER'S EXAM TOPICS IN ANALYSIS

THE FOLLOWING ARE THE BASIC KNOWLEDGE A MASTER'S STUDENT SHOULD HAVE

CONTROL OF, THIS INCLUDES ALL DEFINIITIONS, IMPORTANT THEOREMS (CERTAINLY

NAME BRAND THEOREMS), AND

STANDARD EXAMPLES RELATING TO THIS TOPICS.

TOPOLOGY OF THE REALS

1) Open and closed

2) Accumulation points (limit points, cluster points)

3) Closure of a set

4) Dense sets

5) Compact (and sequentially compact)

6) Connected

7) Completeness Axiom

8) Rationals and Irrationals

9) Countable and uncountable sets

10) Bolzano/Heine Borel etc.

SEQUENCES

1) Convergence

2) Subsequence

3) Cluster points (accumulation points, limit points)

4) Cauchy

5) Monotone sequences

6) Bolzano Theorem

7) Algebra of sequences

LIMITS OF FUNCTIONS

1) Sequence definition of limit

2) Algebra of limits

3) Limits of monotone functions

CONTINUITY

1) Sequences and continuity

2) Algebra of continuous functions

3) Monotone functions

4) Uniform continuity

DERIVATIVES

1) Sequences and derivatives

2) Arithmetic of derivatives

3) Chain rule

4) Mean Value Theorem and Intermediate Value Theorem

5) L'Hospital's Rule and the Inverse Function Theorem

INTEGRATION

1) Riemann sums and the Riemann Integral

2) Classes of integrable functions

3) Fundamental Theorem of Calculus (several forms)

4) Derivatives of integrals

5) Mean Value Theorem and Change of Variable Theorem

INFINITE SERIES

1) Convergence

2) Absolute convergence/ conditional convergence

3) Tests for convergence (Comparison, Ratio, Root tests)

4) Power Series

5) Taylor Series

SEQUENCES AND SERIES OF FUNCTIONS

1) Pointwise and uniform convergence

2) Consequences of Uniform convergence

3) Uniform convergence of power series

SOME KNOWLEDGE OF TOPICS FROM GRADUATE ANALYSIS IS

EXPECTED THOUGH NOT IN AS MUCH DETAIL. FOR EXAMPLE :

1) LEBESGUE MEASURE

2) LEBESGUE INTEGRATION

3) ELEMENTARY THOUGHTS ABOUT GENERAL TOPOLOGY

4) ALGEBRA OF INTEGRALS