Masters & Ph.D. Qualifying Exam
Analysis: Math 825/826
January 22, 2014, 12:00-6:00pm. Avery Hall 15
• 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) The two parts of this problem are unrelated.
(a) Let f : R → R be a differentiable function with f 0 continuous on R. Assume that there
are L, M ∈ R such that lim f (x) = L and that lim f 0 (x) = M . Prove that M = 0.
x→∞
x→∞
(b) Let [a, b] ⊂ R and x0 ∈ (a, b) be given. Set
P0 := {g : [a, b] → R | g is a polynomial satisfying g(x0 ) = 0} .
Suppose that f : [a, b] → R is continuous and satisfies f (x0 ) = 0. Must there be a sequence
of polynomials in P0 that converges uniformly to f on [a, b]? (Justify your answer.)
∞
(2) (a) Produce sequences {an }∞
n=1 and {bn }n=1 of positive real numbers such that
lim inf (an bn ) > (lim inf an )(lim inf bn )
{an }∞
n=1
n→∞
{bn }∞
n=1
(b) Let
and
convergent. Prove that
n→∞
n→∞
be sequences of positive real numbers. Suppose that {an }∞
n=1 is
lim inf (an bn ) = ( lim an )(lim inf bn ).
n→∞
n→∞
n→∞
(3) (a) Is there a function f : R → R, that is differentiable on R, such that
0, if x < 0;
0
f (x) =
1, if x ≥ 1.
(Justify your answer.)
(b) Define f : R → R by
2
x sin sin x1 + sin x + 1,
f (x) :=
1,
if x 6= 0;
if x = 0.
Prove that f is differentiable on R and that f 0 is not continuous at 0.
(4) Let (X, ρ) be a compact metric space, and suppose that f : X → X satisfies
ρ(f (x), f (y)) < ρ(x, y)
for each x, y ∈ X satisfying x 6= y.
(a) Verify that f is uniformly continuous on X.
(b) Prove that there exists a unique x0 ∈ X such that f (x0 ) = x0 .
(5) Suppose that f : R → R has a continuous derivative on [−1, 1] and possesses a second derivative
on (−1, 1). Assume that |f 00 (x)| ≤ 1 for each x ∈ (−1, 1).
(a) Using Taylor’s theorem, verify that
|f (x) + f (−x) − 2f (0)| ≤ x2
for each x ∈ [−1, 1].
(b) Prove that
Z
1
1
f (x) dx − 2f (0) ≤ .
3
−1
(c) Suppose also that |f (x)| ≤ 1 for each x ∈ [−1, 1].
Show that |f 0 (x)| ≤ 2 for all x ∈ [−1, 1].
P
∞
(6) Let {aj }∞
j=1 be a sequence of real numbers such that
j=1 aj is absolutely convergent. For each
n ∈ N define fn : R → R by
n
X
fn (x) :=
aj cos(jx).
j=1
Prove that there is an f : R → R such that {fn }∞
n=1 converges uniformly to f on R.