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.

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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.

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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.

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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.

2