Real Analysis Qualifying Exam
Date: June 06, 2015
Duration: 3 Hours
Instructions: This exam consists of 5 questions. Each question is worth 5 points giving a grand total
of 25 points possible. Please present all of your work in a clear and concise manner and answer each
question as completely as possible. Unsupported work will receive no credit and partially completed work
may receive partial credit. Good luck!
1. Show that the Least Upper Bound Property of the real numbers implies the Cauchy Completeness
Property of the real numbers. That is, show that the property of every bounded set having least upper
bound implies that every Cauchy sequence of real numbers converges in R.
2. (a) Let f : R → R be a continuous real-valued function on R and suppose K ⊂ R is compact. Use an
open cover argument to show that the image f (K) of K under f is compact.
(b) Provide an example (with justification) to show that closure is a necessary condition in part (a). That
is, give an example of a continuous function f that takes a bounded set to a noncompact set.
3. Let X ⊆ R. A real-valued function f : X → R is said to be Lipschitz continuous on X, if there
exists a constant C ≥ 0 such that
|f (x) − f (y)| ≤ C|x − y| for all x, y ∈ X.
(a) Show that f (x) =
√
x is Lipschitz continuous on [1, ∞) but not on [0, ∞).
(b) Show that every Lipschitz continuous function on X ⊂ R is uniformly continuous on X.
sin nx
4. Let a > 0. For each n ∈ N, consider the function fn : R → R defined by fn (x) = √
.
1 + n2
(a) Show that the series of functions
∞
X
fn (x) converges uniformly on [−a, a].
n=1
(b) Show that the series of functions
∞
X
fn (x) is continuously differentiable on [−a, a].
n=1
5. (a) State the definition for a real-valued function f : [a, b] → R to be Riemann integrable on the interval
[a, b].
(b) Let f : [a, b] → R be a continuous function on [a, b]. Use the definition of Riemann integrability to
show that f is Riemann integrable.
Note: If you choose to work with a definition of Riemann integrability different than that stated in part
(a), please provide this alternate definition.