Entrance Exam, Real Analysis
September 1, 2017
Solve exactly 6 out of the 8 problems
1. Prove by de�nition (in ε− δ language) that f(x) =√1 + x2 is uniformly continuous
in (0, 1). Is f(x) uniformly continuous in (1,∞)? Prove your conclusion.
2. Let fn(x) =nx
n+ x.
(a) Find f(x) = limn→∞ fn(x);
(b) Does fn converge to f uniformly (0, 1)? Prove your conclusion.
(c) Does fn converge to f uniformly (1,∞)? Prove your conclusion.
3. If limn→∞
sn ≥ 0, which of the following statements are true? Explain your conclusion.
(a) ∃N > 0 such that ∀n > N , sn ≥ 0.
(b) ∀N > 0, ∃n > N such that sn ≥ 0.
(c) ∃ε > 0 such that ∀N > 0, ∃n > N with sn ≥ 0.
(d) limn→∞
s2n ≤ ( limn→∞
sn)2.
4. Let g(x) be de�ned as follows
g(x) =
∫ ∞−∞
cos(xy3)
1 + y2dy.
(a) Prove g(x) is continuous in (−∞,∞).
(b) Is g(x) uniformly continuous in (−∞,∞)? absolutely continuous in (−∞,∞)?Prove your conclusion.
5. Let {an}∞n=1 be a sequence of real numbers satisfying∑∞
n=1 |an| < ∞. De�ne f by
f(x) =∑∞
n=1 anxn.
(a) Show that f is a function of bounded variation on x ∈ [−1, 1].(b) If a1 6= 0 and an = 0 for n ≥ 2, is f absolutely continuous on x ∈ R. Explain
your answer.
(c) If a1 6= 0, a2 6= 0, and an = 0 for n ≥ 3, is f absolutely continuous on x ∈ R.
Explain your answer.
1
6. For 1 ≤ p <∞ de�ne
‖f‖p = (
∫E|f |p)1/p
and denote Lp(E) to be the set of functions f for which∫E |f |
p <∞.
Either prove the statement or show a counter example.
(a) If E = [0, 1], then there is a constant c > 0 for which
‖f‖1 ≤ c ‖f‖2 for all f ∈ L2(E).
(b) Suppose E = [1,∞). If f is bounded and∫E |f |
2 < ∞, then it belongs to
L1(E).
7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R.
Show that
lim|x|→∞
f(x) = 0.
8. Let {un}∞n=1 be a sequence of Lebesgue measurable functions on [0, 1] and assume
limn→∞ un(x) = 0 a.e. on [0, 1], and also ‖un‖L2[0,1] ≤ 1 for all n. Prove that
limn→∞
‖un‖L1[0,1] = 0.
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Entrance Exam, Real Analysis
April 21, 2017
Solve exactly 6 out of the 8 problems
1. Is f(x) = x3 uniformly continuous in (-5, 1)? in (1,∞)? Prove your conclusion.
2. Let fn(x) =nx
n+ x.
(a) Find f(x) = limn→∞ fn(x);
(b) Does fn converge to f uniformly (0, 1)? Prove your conclusion.
(c) Does fn converge to f uniformly (1,∞)? Prove your conclusion.
3. If limn→∞
sn ≤ 0, which of the following statements are true? Explain your conclusion.
(a) ∃N > 0 such that ∀n > N , sn ≤ 0.
(b) ∀N > 0, ∃n > N such that sn ≤ 0.
(c) ∃ε > 0 such that ∀N > 0, ∃n > N with sn > −ε.(d) lim
n→∞s2n > ( lim
n→∞sn)
2.
4. Let g(x) be de�ned as follows
g(x) =
∫ ∞−∞
cos2(xy)
1 + y2dy.
(a) Prove g(x) is continuous in (−∞,∞).
(b) Is g(x) uniformly continuous in (−∞,∞)? absolutely continuous in (−∞,∞)?Prove your conclusion.
5. Let {an}∞n=1 be a sequence of real numbers satisfying∑∞
n=1 |an| < ∞. De�ne f :[−1, 1]→ R by f(x) =
∑∞n=1 anx
n. Show that f is a function of bounded variation.
6. Suppose that f : [0,∞)→ [0,∞) is measurable and that∫∞0 f(x) <∞. Prove that
limn→∞
∫ ∞0
xnf(x)
1 + xndx =
∫ ∞1
f(x)dx
7. Suppose that f(x) is a uniformly continuous and Lebesgue integrable function on R.
Show that
lim|x|→∞
f(x) = 0.
8. Let {un}∞n=1 be a sequence of Lebesgue measurable functions on [0, 1] and assume
limn→∞ un(x) = 0 a.e. on [0, 1], and also ‖un‖L2[0,1] ≤ 1 for all n. Prove that
limn→∞
‖un‖L1[0,1] = 0.
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Entrance Exam, Real Analysis
August 31, 2016
Solve exactly 6 out of the 8 problems
1. (a) Prove by de�nition that x2 is uniformly continuous in (−7, 2).(b) Prove by de�nition that x2 is not uniformly continuous in [0,∞).
2. If limn→∞
sn = 0, which of the following statements are true? Explain your conclusion.
(a) ∃N > 0 such that ∀n > N , sn ≤ 0.
(b) ∀N > 0, ∃n > N such that sn ≤ 0.
(c) ∀ε > 0, ∃N > 0, such that ∀n > N , sn > −ε.(d) ∃ε > 0 such that ∀N > 0, ∃n > N with sn > −ε.
3. Let f(x) be a function of bounded variation de�ned on [0,1]. Prove that the set ofall discontinuity points of f(x) is countable.
4. Let f(x) be a measurable function de�ned in R and satisfying:
(a) f(0) = 1;
(b) If f(x0) > 0, then ∃δ > 0 such thatf(x) > 0 in (x0 − δ, x0 + δ);
(c) If f(xn) > 0 and xn → a, then f(a) > 0).
Is it true f(x) > 0 ∀x ∈ R? Prove your conclusion.
5. Suppose that {fn} is a sequence of real valued measurable functions de�ned on theinterval [0, 1] and suppose that
limn→∞
fn(x) = f(x) a.e. on [0, 1].
Let p > 1, M > 0, and that‖fn‖Lp(R) ≤M for all n.(a) Prove that f ∈ Lp(R) and that ‖f‖Lp(R) ≤M .
(b) Prove that limn→∞ ‖f − fn‖Lp(R) = 0.
6. Suppose that f ∈ L1(R) is a uniformly continuous function. Show that
lim|x|→∞
f(x) = 0.
1
7. For 1 ≤ p <∞ de�ne
‖f‖p = (
∫E|f |p)1/p
and denote Lp(E) to be the set of functions f for which∫E |f |
p < ∞. For eachstatement below, either prove it or show a counter example.
(a) If E = [0, 1], then there is a constant c ≥ 0 for which
‖f‖2 ≤ c ‖f‖1 for all f ∈ L1(E).
(b) Suppose E = [1,∞). If f is bounded and∫E |f |
2 < ∞, then it belongs toL1(E).
8. (a) For a Lebesgue integrable function f on [0,1] de�ne F (x) =∫ x0 f(t)dt for x ∈
[0, 1]. Prove that F is absolutely continuous on [0,1].
(b) Give an explicit example of a continuous function F of bounded variation on[0,1] that is not absolutely continuous on [0,1]. It is not necessary to verify yourassertion.
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Entrance Exam, Real Analysis
April 21, 2016
Solve exactly 6 out of the 8 problems
1. Let f be a monotone function de�ned on [0,1] with f(0) = 0 and f(1) = 1. Showthat f has at most countable discontinuous points.
2. Answer the following questions and prove your conclusion.
(a) Is x2 uniformly continuous in the open interval (−5, 2)?(b) Is x2 uniformly continuous in [0,∞)?
3. If limn→∞
sn > 0, which of the following statements are true? Prove your conclusion.
(a) ∃N > 0 such that ∀n > N , sn > 0.
(b) ∃ε > 0 such that ∀N > 0, ∃n > N with sn > ε.
(c) ∀ε > 0, ∃N > 0, such that ∀n > N , sn > −ε.(d) lim
n→∞s2n = ( lim
n→∞sn)
2.
4. Let f ∈ L1(−∞,∞), and g(x) be de�ned as follows
g(x) =
ˆ ∞−∞
cos2(xy)f(y)dy.
(a) Prove g(x) is continuous in (−∞,∞).
(b) Is g(x) uniformly continuous in (−∞,∞)? absolutely continuous in (−∞,∞)?Prove your conclusion.
5. Let f and g be real valued measurable functions on [0, 1] with the property that gis di�erentiable at every x on [0, 1] and
g′(x) = (f(x))2.
Show that f∈ L1[0, 1].
6. (a) Let m be the Lebesgue measure on R and let E ∈ R be measurable. Also,suppose f ∈ L1(E) and that f > 0 a.e. on E. Show that
limn→∞
ˆE|f(x)|1/ndx = m(E).
(b) Let E = [a, b], where a and b are real numbers satisfying a < b. With the sameassumptions on f as in (a), show that
limn→∞
(
ˆ b
a|f(x)|1/ndx)n =∞ if b− a > 1
and
limn→∞
(
ˆ b
a|f(x)|1/ndx)n = 0 if b− a < 1.
1
7. Suppose that f : [0,∞)→ [0,∞) is measurable and that´∞0 f(x) <∞. Prove that
limn→∞
ˆ ∞0
xnf(x)
1 + xndx =
ˆ ∞1
f(x)dx
8. Let {un}∞n=1 be a sequence of Lebesgue measurable functions on [0, 1] and assumelimn→∞ un(x) = 0 a.e. on [0, 1], and also ‖un‖L2[0,1] ≤ 1 for all n. Prove that
limn→∞
‖un‖L1[0,1] = 0.
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Entrance Exam, Real Analysis
April 17, 2014
Solve exactly 6 out of the 8 problems.
1. Answer the following questions and prove your conclusion.
• If A is a closed subset of [0,1] and A 6= [0, 1]. Is it possible that mA = 1?
• If B is an open subset of [0,1] and dense in [0, 1]. Is it possible that mB < 1?
2. Let {pk} be all the rational numbers in [0,1], and H(x) be the Heaviside functiondefined as
H(x) =
{1, x > 0,0, x ≤ 0.
Define f(x) in [0,1] as
f(x) =∞∑k=1
(−1)k
2kH(x− pk).
• Prove by definition (in ε− δ language) that f(x) is continuous at all irrationalnumbers, and discontinuous at all rational numbers in [0,1].
• Determine and explain: (i). Is f(x) Riemann integrable in [0,1]? (ii). Is f(x)Lebegue integrable? (iii) Is f(x) of bounded total variation?
3. • Let f be Lebegue integrable in [0,b] with b <∞. Prove
limn→∞
∫ b
0f(x) cosnx dx = 0.
• Is it also true if b =∞? Prove your conclusion.
4. • If f is absolutely continuous on [0,1] and f(x) > 0 on [0,1]. Prove by definitionthat 1/f is absolutely continuous on [0,1].
• Is f(x) =√x absolutely continuous on [0,1]? In [1,∞)? Prove your conclusion.
5. Prove or disprove that for every ε > 0 and for every f ∈ L∞[a, b], where a and b arefinite, there is a g ∈ C[a, b] such that
ess supa≤x≤b|f − g| < ε.
6. Compute the Lebesgue integral∫ 1
0lim infn→∞
xex sin2(πnx) dx.
1
7. Prove or disprove the following statement.
(a) Let g be an integrable function on [0; 1]. Then there is a bounded measurablefunction f such that ∫ 1
0fg = ‖g‖1 ‖f‖∞ .
(b) Let {fn} be a sequence of functions in L1[0, 1], which converge almost every-where to a function f in L1, and suppose that there is a constant M such that‖fn‖1 ≤M all n. Then for each function g in L∞ we have∫ 1
0fg = lim
n→∞
∫ 1
0fng.
8. Let {an}∞n=1 be a sequence of real numbers satisfying∑∞
n=1 |an| < ∞. Define f :[−1, 1]→ R by f(x) =
∑∞n=1 a
2nx
n. Show that f is a function of bounded variationin [−1, 1].
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Entrance Exam, Real Analysis
April 22, 2015
Solve exactly 6 out of the 8 problems.
1. Let A ⊂ R be an open set with the following property: ∀{xn} ⊂ A, ∃ a subsequence{xnk
} such that xnk→ x0 ∈ A.
Which of the following statements is true?mA = 0? mA = ln 2? mA =
√π? mA = ∞? mA is not uniquely determined and
could be any positive number?Prove your conclusion.
2. Answer the following questions and prove your conclusion.
• Let A ⊂ [0, 1] be an open set. If ∃δ < 1 such that mA ≤ δ, is it possible thatA is dense in [0,1]?
• Let B ⊂ [0, 1] and B 6= [0, 1]. If ∀ε > 0, mB > 1 − ε, is it possible that B isclosed?
3. Let f ∈ L1(−∞,∞), and g(x) be defined as follows
g(x) =
ˆ ∞−∞
cos(xy)f(y)dy.
• Prove g(x) is continuous in (−∞,∞).
• Is g(x) uniformly continuous in (−∞,∞)? Prove your conclusion.
4. Let f(x) > 0 and be absolutely continuous on [0,1]. Let g(x) =1√f(x)
. Prove g(x)
is absolutely continuous on [0,1].
5. (a) For a bounded Lebesgue integrable function f on [0,1] define F (x) =´ x0 f(t)dt
for x ∈ [0, 1]. Prove that F is absolutely continuous on [0,1].
(b) Give an explicit example of a continuous function F of bounded variation on[0,1] that is not absolutely continuous on [0,1]. It is not necessary to verify yourassertion.
1
6. (a) Suppose that f ∈ L1[0, 1] and let m denote Lebesgue measure. Prove that forc > 0
m{|f(x)| ≥ c} ≤ 1
c
ˆ 1
0|f(x)|dx.
(b) Suppose that {fn} ⊆ Lp[0, 1] is a sequence of functions satisfying ‖fn‖Lp[0,1] ≤M,where 1 < p <∞. Prove that
limc→∞
ˆ|fn(x)|≥c
|fn(x)|dx = 0,
uniformly in n.
7. Assume that n ≥ 1 is an integer and let f ∈ ∩∞n=1Ln[0, 1]. Prove that if
∞∑n=1
‖f‖Ln[0,1] <∞,
then f = 0 a.e.
8. Suppose that {fn} is a sequence in L1(R) with ‖fn‖L1(R) ≤ 1 for all n and
limn→∞
fn(x) = f(x) a.e.
(a) Prove that f ∈ L1(R) and that ‖f‖L1(R) ≤ 1.
(b) Show thatlimn→∞
(‖f − fn‖L1(R) − ‖fn‖L1(R) + ‖f‖L1(R)) = 0.
Hint: The following inequality might be useful. For any numbers a and b
0 ≤ |a− b| − |a|+ |b| ≤ 2|b|
2
Entrance Exam, Real Analysis
August 27, 2013
Solve exactly 6 out of the 8 problems
1. Given function φ(x) with φ(1) = 1 and φ′(x) = e−x2. The plane curve Γ is de�ned
by the equation
y = φ(1 + xy + y2).
Find the equation for the tangent line to the curve Γ at the point (x, y) = (−1, 1).
2. Does
fn(x) = xn sin
(1− xx
)converge uniformly on (0,1)? Prove your conclusion.
3. Let {xn = pn/qn ∈ Q} be a sequence of rational numbers and xn → α ∈ R\Q (α is
irrational). Prove qn →∞.
4. Let E be a Lebesgue measurable set in R, and fn be a sequence of nonnegative
Lebesgue integrable functions on E. If fn → f a.e. on E, is it always true that
ˆEfdx ≤ limn→∞
ˆEfndx?
Prove your conclusion.
5. Let a and b are real numbers satisfying a < b. Prove or disprove the following.
(a) If f(x) is a di�erentiable function on a < x < b, it is absolutely continuous on
a < x < b.
(b) If f(x) is absolutely continuous on a ≤ x ≤ b, it is Lipschitz on a ≤ x ≤ b.
6. A family F of measurable functions on E of �nite measure is said to be uniformly
integrable over E provided that for each ε > 0, there is a δ > 0 such that for each
f ∈ F and for each measurable set A ⊆ E with mA < δ,´A |f | < ε. Either prove or
disprove the following statements.
(a) If G is the family of measurable functions f on [0, 1], each of which is integrable
over [0, 1] and has´ 10 |f | < 1, then G is uniformly integrable over [0, 1].
(b) If {fn} is a sequence of nonnegative, measurable, and integrable functions that
converges pointwise a.e. on E to h = 0, then {fn} is uniformly integrable.
1
7. For 1 ≤ p <∞ de�ne
‖f‖p = (
ˆE|f |p)1/p
and denote Lp(E) to be the set of functions f for which´E |f |
p < ∞. Either prove
the statement or show a counter example.
(a) If E = [0, 1], then there is a constant c ≥ 0 for which
‖f‖2 ≤ c ‖f‖1 for all f ∈ L1(E).
(b) Suppose E = [1,∞). If f is bounded and´E |f |
2 < ∞, then it belongs to
L1(E).
8. Prove or disprove that for every ε > 0 and for every f ∈ L∞[a, b], where a and b are�nite, there is a g ∈ C[a, b] such that
ess supa≤x≤b|f − g| < ε.
2
Entrance Exam, Real Analysis
April 22, 2013
Solve exactly 6 out of the 8 problems
1. Given function φ(x) with φ(1) = 5 and φ′(x) = e−x2. The plane curve Γ is de�ned
by the equation
y = φ(1 + sin 3(xy)).
Find the equation for the tangent line to the curve Γ at the point (x, y) = (0, 5).
2. Does
fn(x) = xn sin(
1− xx
)converge uniformly on (0,1)? Prove your conclusion.
3. Evaluate the integral
I =ˆ 1
0dy
ˆ 1
ye−x2
dx.
4. Let f , g be absolutely continuous in (0,1). Is it true that the product fg is also
absolutely continuous in (0,1)? Prove your conclusion.
5. Let {fn} be a sequence of nonnegative integrable functions on E. For each of (a),
(b), and (c) either prove the statement or show a counter example.
(a) If limn→∞´E fn = 0, then {fn} → 0 in measure.
(b) If {fn} → 0 in measure, then limn→∞´E fn = 0.
(c) If {fn} → 0 almost everywhere on E, then limn→∞´E fn = 0.
6. For f in C(E), where C(E) is the set of continuous functions on E, de�ne
‖f‖p = (ˆ
E|f |p)1/p, ‖f‖sup = sup
x∈E|f(x)|.
Here 1 ≤ p <∞. For each of (a), (b), and (c) either prove the statement or show a
counter example.
(a) If E = [0, 1] and 1 ≤ p < q <∞, then there is a constant c ≥ 0 for which
‖f‖p ≤ c ‖f‖q for all f ∈ C(E).
(b) If E = [0, 1], then there is a constant c ≥ 0 for which
‖f‖sup ≤ c ‖f‖1 for all f ∈ C(E).
(c) Suppose E = [1,∞). If f ∈ C(E) is bounded and´E |f |
2 <∞, then it belongs
to L1(E).
1
7. Prove or disprove that for every ε > 0 and for every f ∈ L∞[a, b], there is a g ∈ C[a, b]such that
ess supa≤x≤b|f − g| < ε.
8. (a) Suppose E ⊆ R has Lebesgue measure zero. Prove that if f : R → R is
increasing and absolutely continuous, then f(E) has Lebesgue measure zero. Here,
f(E) = {y | y = f(x) for some member x of E}.
(b) Would the same conclusion hold if we remove �increasing� in the assumption on
f? In other words, if f : R → R is absolutely continuous, then would f(E) have
Lebesgue measure zero? Explain your reasoning.
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