Analysis Qualifying Exam
Spring 2010
This exam has five (5) questions. Please answer each part as completely as
possible. Unsupported work will receive no credit, and partially completed work
may receive partial credit. Each question is worth five (5) points, for a grand total
of 25 points possible. Good luck to you all!
1. Let K be a compact subset of R and let f be a continuous function from K into
R. Use the open cover definition of compactness to show that f (K) is compact.
2. Let {an } be a sequence of real numbers which converges to a. Prove
a1 + a2 + · · · + an
= a.
n→∞
n
lim
Hint: If 1 ≤ j ≤ n and b1 , b2 , . . . , bn are real numbers, then b1 + b2 + · · · + bn =
(b1 + b2 + · · · + bj ) + (bj+1 + bj+2 + · · · + bn ).
3. Let f : [a, b] → R be a continuous function. Suppose f is differentiable on (a, b)
and f 0 (x) > 0 for all x ∈ (a, b).
a) Show f has an inverse defined on [f (a), f (b)].
0
b) Prove that f −1 is differentiable on (f (a), f (b)) and (f −1 (x)) =
1
f 0 (f −1 (x))
4. a) State the definition for a function f to be Riemann integrable on an interval
[a, b].
b) Let
f (x) =
−1 if x ∈ Q
1
if x ∈ R\Q
Use the defintion stated in part (a) to prove f is not Riemann integrable on [0, 1].
c) Let
g(x) =
x
if 0 ≤ x < 1
x + 1 if 1 ≤ x ≤ 2
Use the defintion stated in part (a) to prove g is Riemann integrable on [0, 2].
5. a) Let {fk } be a sequence of real-valued functions defined on a set E ⊂ R. State
∞
P
the definition for the series
fk (x) to converges uniformly on E.
k=0
b) Use the definition stated in (a) to prove that
∞
P
xk converges uniformly on any
k=0
closed interval [a, b] such that [a, b] ⊂ (−1, 1).
c) Use the definition stated in (a) to prove that
∞
P
k=0
on (−1, 1).
xk does not converges uniformly