Analysis Qualifying Exam
September 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!
1. Find
5x2 − 1
x→1 2x + 3
and use the − δ definition of a limit to prove your answer is correct.
lim
2. Suppose f : R → R is differentiable and f 0 is uniformly continuous on R. Define
1
gn (x) = n f (x + ) − f (x)
n
for all natural numbers n. Prove that the sequence gn converges uniformly to f 0 on
R.
3. (a) Assume f is a bounded real-valued function defined on an interval [a, b]. Define
what it means for f to be Riemann integrable on [a, b] in terms of Riemann sums.
(b) Define a function f on the interval [0, 1] by

T
 x for x ∈ Q [0, 1],
f (x) =
 0 for x ∈ (0, 1), x ∈
/ Q.
Is f Riemann integrable on [0, 1]? Prove or disprove your answer directly from the
definition that you gave in part (a).
4. (a) Prove that the series
∞
X
n=1
sin
x
converges uniformly on any closed bounded
n2
interval [a, b].
∞
X
x
(b) Prove that
sin 2 does not converge uniformly on (−∞, ∞).
n
n=1
5. Let S be a compact subset
T of the real numbers and let T be a closed set of real
numbers. Assume that S T = ∅. Prove that there is a number δ > 0 such that
|s − t| > δ for every s ∈ S and every t ∈ T.