Masters and Ph.D. Qualifying Exam
Analysis: Math 825/826
Thursday, June 5, 2014, 12:00 - 6:00p.m.
• Work 5 out of 6 problems. • Each problem is worth 20 points. • Write on one side of the paper only and hand your work in order.
• Do not interpret a problem in such a way that it becomes trivial.
(1) Define α : [−1, 1] → R by
(
α(x) :=
−1, x ∈ [−1, 0];
1,
x ∈ (0, 1].
Let f : [−1, 1] → R be a function that is uniformly bounded on [−1, 1] and continuous at
x = 0, but not necessarily continuous for x 6= 0. Prove that f is Riemann-Stieltjes integrable
with respect to α over [−1, 1] and that
Z 1
f (x) dα(x) = 2f (0).
−1
(2) (a) State the Weierstrass approximation theorem.
(b) Let [a, b] ⊂ R be given. Suppose that f : [a, b] → R is continuous and that
Z b
f (x)xn dx = 0 for each n = 0, 1, 2, 3, . . . .
a
Prove that f (x) = 0 for all x ∈ [a, b].
(3) Suppose that {fn }∞
n=1 is a sequence of continuous functions fn : [0, 1] → R with the following
property: there are numbers 0 < λ ≤ Λ such that for all integers n > 0
λxn ≤ fn (x) ≤ Λxn for each x ∈ [0, 1].
∞
X
fn (x)(1 − x) converges pointwise but not uniformly on [0, 1].
(a) Prove that the series
n=1
∞
X
(b) Prove that the series
(−1)n fn (x)(1 − x) converges uniformly on [0, 1].
n=1
(4) Fix x0 ∈ R. Suppose that f : R → R is infinitely differentiable; i.e. has continuous derivatives
on R of all orders. Let k ∈ {0, 1, 2, . . . } be given.
(a) Provide the formula for Pk : R → R, the k-th order Taylor polynomial for f centered at
x0 .
(b) Suppose that Q : R → R is a polynomial of degree k satisfying
f (x) − Q(x)
lim
= 0.
x→x0 (x − x0 )k
Prove that Q = Pk .
(5) Let (X, ρ) be a metric space.
∞
(a) Suppose that {xn }∞
n=1 and {yn }n=1 are sequences from X such that lim xn = x0 and
n→∞
lim yn = y0 for some x0 , y0 ∈ X. Argue that
n→∞
lim ρ(xn , yn ) = ρ(x0 , y0 ).
n→∞
(For this part do not use, without proof, the fact that ρ is continuous.)
(b) Under the assumption that (X, ρ) is compact, verify that there exist a, b ∈ X such that
ρ(a, b) = sup{ρ(x, y) : x, y ∈ X}.
(For this part you may assume, without proof, that ρ is continuous.)
(6) (a) Let {an }∞
n=1 be a bounded sequence from R. State the definitions of lim sup an and
n→∞
lim inf an .
n→∞
(b) Let {an }∞
n=1 be an enumeration of the rational numbers in (0, 1). Compute, with justification, the numerical values of lim sup an and lim inf an .
n→∞
n→∞