Masters and Ph.D. Qualifying Exam
Analysis: Math 825/826
Wednesday, January 27, 2017, 2:30 – 8:30p.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) Let (X, ρ) be a metric space.
(a) State the definition for a compact set K ⊆ X.
(b) Suppose that {xj }∞
j=1 is a sequence in X that converges to x ∈ X. Verify that the set
K := {x} ∪ {xj : j ∈ N} is compact.
(2) Define f : [0, 1] → [−1, 1] by
x sin x1 , 0 < x ≤ 1;
f (x) :=
0,
x = 0.
(a) Determine, with justification, whether f is of bounded variation on the interval [0, 1].
(b) Determine, with justification, whether f is continuous on the interval [0, 1].
(3) Let f : [0, 1] → R be a non-decreasing function. Prove that limx→c− f (x) exists for each
c ∈ (0, 1] and that limx→c+ f (x) exists for each c ∈ [0, 1).
(4) Let f : R → R be a continuous function on R. Let c ∈ R be given, and suppose that f has the
following property: there is an L ∈ R such that for each ε > 0 there is a δ > 0 such that
f (r) − f (c)
− L < ε, whenever r ∈ Q and 0 < |r − c| < δ.
r−c
Prove that f is differentiable at c and that f 0 (c) = L.
(5) Let f : (0, 1) → R be a continuous function. Fix c ∈ (0, 1), and suppose that f is differentiable
on (0, 1)\{c}. Assuming that limx→c f 0 (x) exists, prove that fPis differentiable at c. P
∞
∞
2
(6) (a) Let {xj }∞
of real numbers such that
j=1 be a sequence
j=1 xj
j=1 xj converges, but
P∞
diverges. Argue that j=1 xj must converge conditionally.
P
P∞
∞
xj and ∞
(b) Let {xj }∞
j=1 and {yj }j=1 be sequences of real numbers such that
j=1 yj
j=1
P∞ p
are both absolutely convergent. Prove that the series
|xj yj | is also absolutely
j=1
convergent.