Comprehensive Exam in Analysis
June 6, 2008
There are 7 problems on this exam. The best 5 will be taken for the final score.
1. Let f be a real valued function defined on the real numbers
(i) Give the definition for f to be uniformly continuous on R
(ii) Prove that: if, for some constant M > 0, f satisfies
If(x) - f(y)1 ::; Mix then
f
yl
for all
X,
L~.
y E ]R,
is uniformly continuous on R
2. Give an example of a real valued function f and a set S ~ ]R such that
is continuous on S but not uniformly continuous on S. Prove all your claims.
f
g(x)-cosx if X :f 0 and f(x) = a if ,x = 0, where g"(x) exx
ists and is continuous for all x, and wher 9(0) = 1,9'(0) = 2, and g"(O) = 4.
Justify all steps in the following.
(i) Make the value a such that f(x) is continuous at x = O.
(ii) Establish the value 1'(0) directly; do not assume continuity of 1'(x) at x = O.
(iii) Show that limx~o 1'(x) exists and equals the value 1'(0) found in part (ii).
3. Let f(x) =
4. In each case determine whether the given sequence of functions fn(x) con­
verges uniformly or not. Justify your answers.
(i) fn(x) =
on R
(ii) fn(x) = L:~=l x k on [-1/2,1/2].
sfi
5. In each case determine whether the given series of numbers converges or
diverges. Justify your answers.
.)
(I
",00
Inn
Ln=2 --:;;:T'
( .. )
II
",00
Ln=l
3
(_1)nn
(1.1)" '
6. Assume that the sequence of real valued continuous functions
uniformly to Jon [0, I].
(i) Prove that f is continuous.
(ii) Prove that
lim
n~oo
Jot
fn dx =
Jot
in
converges
fdx.
7. (i) Let A = {(x,x 2 ) E IR.2: 0::; x::; I}. Show that A is closed in]R2 equipped
with the Euclidean distance.
(ii) Let Ai be the subset of A as follows: Al = {(x,x 2 ) E A : x is rational},
Show that Al is not closed in ]R2.
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