Real Analysis Problems
Cristian E. Gutirrez
e
September 14, 2009
1
1 CONTINUITY
1
Continuity
Problem 1.1 Let rn be the sequence of rational numbers and
1
.
2n
f (x) =
cfw_n:rn <x
Prove that
1. f is continuous on the irrationals.
2. f is discontinuous on t
Real Analysis Qualifying Exam
Department of Mathematics, Temple University
January 14, 2005
All integrals are Lebesgue integrals; do not use the Riemann integral or use any facts relating
to it. Justify your reasoning carefully and clearly.
Part I
Please
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
January 1996
Part I. Do three (3) of these problems.
I.1. Let cfw_an be a sequence of real numbers with the following property: there is a constant
0 < K < 1 such that
|an+2 an+1 | K|an+1 an | for all
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
Spring 1999
Justify carefully all reasoning.
Part I. Do three (3) of these problems.
I.1.
Dene the Lebesgue measure |A| of a set A R. Show that
|A| = inf
n=1
diam(An ) : A
An , An arbitrary .
n=1
Here
Real Analysis Ph.D. Qualifying Exam
Temple University
August 16, 2013
Justify your answers thoroughly.
You are allowed to rely on a previous part of a multi-part problem even if you do not
work out the previous part.
Notation: R and N denote the set of
Real Analysis Ph.D. Qualifying Exam
Temple University
August, 2014
Justify your answers thoroughly.
You are allowed to rely on a previous part of a multi-part problem even if you do not
work out the previous part.
For any theorem that you wish to cite,
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
Spring 1994
Part I. Do three (3) of these problems.
I.1. Let
fn (x) =
n
0
1
if 0 < x < n
1
if n < x < 1.
(a) Find limn fn (x).
1
(b) Find limn 0 fn (x) dx.
(c) Is there a function g(x) L1 (0, 1) such t
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
January 1995
Part I. Do three (3) of these problems.
I.1. Give an example of a closed set which contains no interval and has Lebesgue measure
equal to 2.
I.2. State each of the following inequalities.
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
January 2002
Part I. Do three (3) of these problems.
I.1. Show that
2n
lim
n
k=0
k2
1
k
= ln 5.
2
+n
2
I.2. Let cfw_xk be a sequence of real numbers satisfying
k=1
x 1 + + xn
(a) limn
= L, and
n
1 n
k
Mathematics Real Analysis Ph.D. Qualifying Exam
Friday, January 14, 2006
All functions on Rd are assumed Lebesgue measurable, all integrals are against Lebesgue measure, and
L = Lp (Rd ). You may not use or refer to the Riemann integral in any of your ans
Real Analysis Ph.D. Qualifying Exam
Temple University
January, 2014
Justify your answers thoroughly.
You are allowed to rely on a previous part of a multi-part problem even if you do not
work out the previous part.
Notation: R and N denote the set of r
Real Analysis Ph.D. Qualifying Exam
Temple University
January 2013
Justify your answers thoroughly.
You are allowed to rely on a previous part of a multi-part problem even if you do not work
out the previous part.
Notation: R and N denote the set of re
Real Analysis Ph.D. Qualifying Exam
Temple University
January 13, 2012
Part I. (Do 3 problems)
k=1
1. Let xk be a sequence in a metric space (X, d) such that
is a Cauchy sequence.
2. Prove that the function
+
F(x) =
0
d(xk , xk+1 ) < . Prove that xk
cos(x
Real Analysis Ph.D. Qualifying Exam
Temple University
January 14, 2011
Part I. (Select 3 questions.)
1. Let v be any nonnegative function dened in all Rn and for each integer j > 0, let B j
denote the ball with center 0 and radius j. Prove that
inf v(x) i
Real Analysis Ph.D. Qualifying Exam
Temple University
January 15, 2010
Part I. (Select 3 questions.)
1. Let f : [a, b] R be a bounded function and set
M = sup f (x),
m = inf f (x),
[a,b]
[a,b]
M = sup | f (x)|,
m = inf | f (x)|.
[a,b]
[a,b]
Prove that M m
Mathematics Real Analysis Ph.D. Qualifying Exam
January 16, 2009
PART I (select three questions)
1. Let f : R R be C n (R) for some n 0. Prove that if f (k) (0) = 0, for all 0 k n, then
f (k) (x)
0 as x 0, for all 0 k n.
|x|nk
2. Prove that on C[0, 1] th
Mathematics Real Analysis Ph.D. Qualifying Exam
Temple University
January 12, 2007
December 20, 2006
All functions on Rd are assumed Lebesgue measurable and all integrals are against Lebesgue measure.
You may not use or refer to the Riemann integral in an
Mathematics Real Analysis Ph.D. Qualifying Exam
Temple University
January 18, 2008
All functions on Rd are assumed Lebesgue measurable and all integrals are against Lebesgue
measure. Justify your answers.
Part I. (Select 3 questions.)
1 22
1. Let fn (x) =
Real Analysis Ph.D. Qualifying Exam
Temple University
August 24, 2012
Part I. (Do 3 problems)
1. Let f C2 (a, b). Prove that
lim
h0
f (x + h) + f (x h) 2 f (x)
= f (x)
h2
for each x (a, b).
2. Let fn (x) =
x2
nx
, x R. Show that
+ n2
(a) fn does not conve
Real Analysis Ph.D. Qualifying Exam
Temple University
August 26, 2011
Part I. (Do 3 problems)
1. Let f C1 (R) with | f (x)| M for all x. Prove that
(a) if g BV[a, b], then the composition f g is of bounded variation in [a, b].
(b) if g is absolutely conti
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
August 1996
Part I. Do three (3) of these problems.
I.1. Determine all the values of p for which the limit
sin(| sin x|p )
x0
x
lim
exists and calculate its value. Justify your answer.
I.2. Show that t
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
Fall 1994
Part I. Do three (3) of these problems.
I.1. (a) Give an example of a function f (x) such that limm
m
but limm 0 |f (x)| dx does not exist.
m
0
f (x) dx exists,
(b) Give an example of a funct
Real Analysis II, Math 8042, Prof. Guti rrez
e
Semicontinuity, maximal function
Week of January 17, 2012
Notation.
Let Rn be open. The function f : R is lower (upper) semicontinuous if f (z)
lim inf xz f ( x) = lim0 inf | xz|< f ( x) ( f (z) lim sup xz
Real Analysis II, Math 8042, Prof. Guti rrez
e
p
Absolute continuity and L
Week of January 30, 2012
1. Show that if f and g are absolutely continuous functions in [a, b] and f ( x) = g ( x) a.e., then
f ( x) g( x) = constant, for each x [a, b].
2. Show th
Real Analysis II, Prof. Guti rrez
e
Abstract measures II
Week of March 27, 2012
Assume (S, , ) is a measure space, i.e., is a -algebra of sets in S and is a measure on .
1. Let : R be an additive set function. Prove that
N
N
V (E ) = sup |(Fi )| : cfw_Fi
Real Analysis II, Prof. Guti rrez
e
Abstract measures I
Week of March 13, 2012
Assume (S, , ) is a measure space, i.e., is a -algebra of sets in S and is a measure on .
1. Prove that the following triplets (S, , ) are measure spaces.
1. S is an innite set
Real Analysis II, Math 8042
Prof. Guti rrez
e
Miscellaneous Problems
Spring 2012
1. Let a < b, c < d. Dene
ax sin2 1 + bx cos2 1 , for x > 0
x
x
f ( x) = 0,
for x = 0
cx sin2 1 + dx cos2 1 , for x > 0.
x
x
Calculate D f , D+ f , D f , D+ f at x = 0.
2. Le
PH.D. COMPREHENSIVE EXAMINATION
REAL ANALYSIS SECTION
Fall 1995
Part I. Do three (3) of these problems.
I.1. Let be the Lebesgue measure on R. Let (x) = x2 . Dene a measure by
(A) = (1 (A), for all Lebesgue measurable sets A.
Find the Radon-Nikodym deriva