Analysis Qualifying Exam
June 2, 2012
This exam has five (5) questions. Please answer each question 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!
2
x
for all n ∈ N. Show that the sequence {fn }
1. Let fn : [0, 1] → R be defined by fn (x) = 2n
e2nx
converges uniformly on [, 1] for all > 0. Does the sequence {fn } converge uniformly on
[0, 1]?
2. (a) Define what it means for a real-valued bounded function to be Riemann Integrable
on an interval [a, c].
(b) Let g be a real-valued continuous function on the interval [a, b] and let h be a real-valued
continuous function on the interval [b, c]. Assume g(b) > h(b). Define the function f defined
on [a, c] by
f (x) =

 g(x) x ∈ [a, b],

h(x) x ∈ (b, c].
Use the Riemann sum definition of Riemann integrable to rigorously prove that f is Riemann
integrable on [a, c]. Do not simply quote a theorem.
3. Define
f (x) =
∞
X
arctan(nx)
n=1
n2
.
Prove that the series defining f converges for each x ∈ R and that f is continuous on R.
You may use elementary properties of arctan(x) without proof.
4. Let f be differentiable on (0, 1) and continuous on [0, 1] with f (0) = f (1) = 0. Prove that
for every real number a there exists a real number c ∈ (0, 1) such that f 0 (c) = af (c).
Hint: Consider g(x)f (x) for an appropriate g(x).
5. Assume f is a real-valued continuous function defined on a compact subset K of R. Prove
that f is uniformly continuous on R.