Material for Preliminary Exam A - Analysis Section

 

 

The following list gives topics on which the Preliminary Examination A in Analysis will be based. Math 5307 covers many, but not necessarily all, of these topics. Students (even those who have taken Math 5307) are advised to prepare for the examination using many resources, including (but not limited to) the books suggested below.

 

The number of lectures per section is only a recommendation and is based on 1 hour 20 minute lecture time. The exact amount of time spent on each topic will be decided by the instructor.

 

1. The Real Number System (1 lecture)

a. rationals, irrationals, and their properties

b. upper and lower bounds, supremum and infimum, maximum and minimum

c. completeness axiom

d. triangle inequality and Cauchy-Schwarz inequality

 

2. Basic Notions of Set Theory (2 lectures)

a. Cartesian product of sets, ordered pairs

b. relations and functions, sequences, composition

c. injectivity, surjectivity, bijectivity, inverses

d. similar sets, (in)finite sets, (un)countable sets, examples

e. set algebra

 

3. Elements of Topology (4 lectures)

a. metric spaces

b. open and closed sets, compactness

c. adherent points, accumulation points, interior points, boundary points

d. Bolzano-Weierstrass theorem

e. Cantor intersection theorem

f. Lindelöf covering theorem

g. Heine-Borel covering theorem

 

4. Limits and Continuity (6 lectures)

a. convergent sequences in a metric space, Cauchy sequences, completeness

b. limits of functions, continuity, uniform continuity

c. intermediate-value theorem for continuous functions,  discontinuities of real-valued functions

d. continuity and inverse images of open or closed sets

e. homeomorphisms

f. connectedness and arcwise connectedness, components of a metric space

g. fixed-point theorem for contractions

 

 5.  Derivatives (3 lectures)

a. analytic and geometric notions of derivative, properties of derivative

b. one-sided derivatives and infinite derivatives

c. local extrema

d. mean-value theorem for derivatives,  intermediate-value theorem for derivatives

e. Taylor's formula with remainder

 

6.  Functions of Bounded Variation and Rectifiable Curves (3 lectures)

a. monotonic functions

b. functions of bounded variation

c. total variation and its properties

d. curves, paths, arc lengths, and their properties

 

7.  Riemann-Stieltjes Integral (RSI) (10 lectures)

a. definition and linear properties of RSI

b. integration by parts and change of variable in RSI

c. reduction to a Riemann integral

d. step functions as integrators and reduction of RSI to a finite sum, Euler’s summation formula

e. monotonically increasing integrators, upper and lower integrals, their properties

f. Riemann’s condition

g. comparison theorems

h. integrators of bounded variation

i. sufficient and necessary conditions for the existence of RSI

j. mean-value theorems for RSI

k. integral as a function of interval, first and second fundamental theorems of integral calculus

l. RSI depending on a parameter,

m. differentiation under the integral sign, interchanging the order of integration

n. Lebesgue criterion for existence of Riemann integral

o. change of variable in a Riemann integral, second mean-value theorem for Riemann integrals

 

 

Some helpful books include:

 

[1] Tom M. Apostol, “Mathematical Analysis” (second edition), Addison-Wesley, 1974

 

[2] Richard A. Goldberg, “Methods of Real Analysis” (second edition), Wiley, 1976

 

[3] Walter Rudin, “Principles of Mathematical Analysis” (third edition), McGraw-Hill, 1976