PH.D. PRELIMINARY EXAMINATION IN ANALYSIS
ANALYSIS FACULTY GROUP
Number
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This exam lasts four hours.
No document, computer, calculator, cell phone and any other aid is allowed.
Each problem is worth ten points.
Please indicate which problems you wish to be graded.
– Only marked problems will be graded,
– Do not mark more than eight problems.
– You are free to choose any eight problems you wish.
• A score of 60 would ensure a pass to this exam.
• Your work will be assessed on its quality and rigor.
G OOD L UCK !
1. R EAL A NALYSIS
(1) Let a < b ∈ R. Let f : [ a, b] → [0, ∞) be a continuous nonnegative function. Show
that:
s
lim
n→∞
n
Z b
a
( f ( x ))n dx = sup{ f ( x ) : x ∈ [ a, b]}.
(2) Let f : [0, 1] → R be a Riemann integrable function which is also continuous at 0,
and with f (0) = 0. Show that:
lim x
x →0+
Z 1
f (t)
t2
x
Date: October 6, 2015.
1
dt = 0.
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ANALYSIS FACULTY GROUP
(3) Does the series:
∑
√
1
n 1 − cos
n
n ∈N
converge? Prove whatever series convergence or divergence tests you use.
(4) Let f : R → R be continuous. For all n ∈ N, n > 0 and x ∈ R, we let:
1 n −1
f n (x) = ∑ f
n k =0
k
x+
n
.
Prove that on any compact interval [ a, b] (with a < b ∈ R), the sequence ( f n )n∈N
converges uniformly, and provide its limit in terms of f .
(5) Prove that an increasing function f : [0, 1] → R is the pointwise limit of a sequence
of continuous functions over [0, 1]. Hint: Prove that for any interval I of [0, 1], the
function:
(
1 if x ∈ I
x ∈ [0, 1] 7→
0 otherwise
is a pointwise limit of continuous functions on [0, 1].
2. M ETRIC S PACES
(1) Let ( E, d) be a compact metric space and let f : E → R be a continuous function.
Prove that f is uniformly continuous on E.
(2) Let ( E, d) be a metric space. For any A ⊆ E, let:
diam ( A) = sup{d( x, y) : x, y ∈ A}.
Show that ( E, d) is complete if, and only if, the following property holds: if
( Hn )n∈N is a sequence of nonempty closed subsets of ( E, d) such that:
lim diam ( Hn ) = 0,
n→∞
and
Hn+1 ⊆ Hn for all n ∈ N,
then there exists x ∈ E such that
T
n ∈N
Hn = { x }.
(3) Prove that Q is not the intersection of a countable collection of open subsets of R.
PH.D. PRELIMINARY EXAMINATION IN ANALYSIS
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3. T OPOLOGY
(1) Let ( E, τ ) be a topological space. Let A ⊆ E. Prove that:
{ x ∈ E : ∀V ∈ τ
x ∈ V =⇒ V ∩ A 6= ∅} .
is the closure of A, i.e. the smallest closed set in τ containing A. Let A0 be the set:
A 0 = { x ∈ E : ∀V ∈ τ
x ∈ V =⇒ V ∩ ( A \ { x }) 6= ∅}
i.e. A0 is the set of limit points of A. Prove that if τ is T1, then:
A 0 = { x ∈ E : ∀V ∈ τ
x ∈ V =⇒ V ∩ A is infinite} .
(2) Let ( E, τE ) and ( F, τF ) be two compact Hausdorff spaces. Let f : E → F. Prove the
following assertions are equivalent:
(a) f is continuous,
(b) the graph:
Graph( f ) = {( x, f ( x )) : x ∈ E}
is closed in ( E × F, τE ⊗ τF ).
(3) Let E be the set of all functions from [0, 1) to {0, 1}, endowed with the topology τ
of pointwise convergence, where {0, 1} is endowed with the discrete topology. Is
there a metric on E which induces the topology τ?
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D EPARTMENT OF M ATHEMATICS , U NIVERSITY OF D ENVER , D ENVER CO 80208