Topics for the Ph.D. Preliminary Examination
ANALYSIS
Fundamentals of Analysis (MAT 601-602)
The basic material is contained in Rudin's "Principles of Mathematical Analysis" 3rd
Edition, Chapters 1 through 9. Important topics are:
1. Properties of real an
January 2010 Preliminary Exam in Analysis.
1. Let X be a connected metric space. Given two points p, q X and a number
prove that there exist an integer n
> 0,
0 and points a0 , a1 , . . . , an X such that a0 = p,
an = q , and
d(aj , aj 1 ) <
for all j = 1
August 2010 Preliminary Exam in Analysis
1. Suppose that f : R R is a function such that f (f (x) = x for all x R. Prove that
there exists an irrational number t such that f (t) is also irrational.
2. Find three subsets A, B, C of the real line R such tha
Analysis preliminary exam
Jan. 8, 2009
1. Let C be the standard Cantor set on the interval [0, 1] and let A = C c be its complement on the real line. Identify the set of all limit points A of A, explaining your answer.
2. (a) Prove
n
k=
k=1
n(n + 1)
2
(b)
Analysis Preliminary Exam
August 24, 2009
1. If F1 and F2 are closed subsets of R1 and dist(F1 , F2 ) = 0 then F1 F2 = . Prove or
give a counterexample.
2. Newtons method for nding zeroes of a function f : R1 R1 is based on the recursion
formula
f (xn )
x
Analysis Preliminary Exam
August, 2008
1. Let f : R2 R be given by the formula
x2 y
x2 + y 2
f (x, y ) =
0
if (x, y ) = (0, 0)
if (x, y ) = (0, 0).
(a) Show that f is continuous at (0, 0).
(b) Prove that the rst order partial derivatives of f at (0, 0) ex
Preliminary Exam Jan 2007
1. Let X be a metric space and let Aj be subsets of X , j = 1, 2, . . . . For each of the following statements, prove it or give a counterexample (the means limit points):
(i) (A1 A2 ) A1 A2
(ii)
j =1
Aj
Aj
j =1
2. Prove that th