TABLE OF CONTENTS · Real Analysis Test 1 ... Countable sets, Sequences and Series, Point-set topology, Continuity, Uniform Continuity, Differentiability, Uniform Convergence,Riemann

TABLE OF CONTENTS

Real Analysis Test 1……………………………………………….……. [#]

Topics: Set theory, Countable Sets, Sequences and Series.


Topics: Continuity, Uniform Continuity, Differentiability.


Topics: Uniform Convergence, Riemann Integration.


Topics: Metric Spaces, Functions of Several Variables.


Topics: : Set theory, Countable sets, Sequences and Series, Point-set topology,

Continuity, Uniform Continuity, Differentiability, Uniform Convergence,Riemann

Integration, Metric Spaces..

Mar 30, 2019

12 thousand via teaching, social

social m

Real Analysis Test-1

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“Be transparent. Let’s build a community that allows hard questions andhonest conversations so we can stir up transformation in one another.” -Germany Kent

Time duration: 80 minutesMax. points: 99 -Praveen Chhikara

Topics: Set theory, Countable Sets, Sequences and Series

Instructions:

1. There are two sections, first of “Single-correct questions” and secondof “Multi-correct questions”.

2. Each question of the first section has one correct option. One questioncarries 3 marks, and a wrong choice leads to NEGATIVE 0.75 marks.

3. Each question of the second section has one or more correct option(s).One question carries 4.75 marks, and there is no negative marking inthis section.

1 Single-correct Questions

1. Which of the following is a one-to-one function f : R→ R?

(a) f(x) = |x|.(b) f(x) = cos x2.

(c) f(x) = x2

x2+1.

(d) f(x) = x|x|x2+1

.

2. The sequence{n. n

√n∏k=1

sin( 1k)}

(a) is unbounded.

(b) is bounded but not convergent.

(c) is convergent and its limit is e.

(d) is convergent and its limit is 1/e.

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3. Consider a sequence (an) in R. Then

(a) if all convergent subsequences of (an) converge to 0, then (an)→ 0.

(b) if all monotone subsequences of (an) are convergent, then (an)must be bounded.

(c) (an) is divergent if, and only if all of its subsequences are divergent.

(d) If (an) diverges, then all of its subsequences diverge.

4. Which of the following is an incorrect statement?

(a) If∞∑n=1

an is a convergent series of positive terms, then∞∑n=1

a3n must

be convergent.

(b) If∞∑n=1

an is a convergent series of positive terms, then∞∑n=1

√anan+1

is convergent.

(c) If∞∑n=0

an and∞∑n=0

bn are convergent series, then so is∞∑k=0

ck, where

ck =k∑l=0

albk−l.

(d) If∞∑n=1

an and∞∑n=1

bn are convergent series of positive terms, then

so is∞∑n=1

anbn.

5. A sequence (an) of real numbers is said to be of bounded variation if

the series∞∑n=2

|an − an−1| converges. Then

(a) every convergent sequence is of bounded variation.

(b) every monotone convergent sequence is of bounded variation.

(c) if (an) and (bn) are of bounded variation, then (an − bn) may notbe of bounded variation.

(d) if (an) and (bn) are of bounded variation, then (anbn) may not beof bounded variation.

6. Let V be a vector space of all bounded real sequences x = (xn)whichconverge to zero. Define‖ x ‖1= sup{| xn |: n ∈ N} and ‖ x ‖2= inf{| xn |: n ∈ N}. Then

(a) if x 6= y, then ‖ x− y ‖1 may be zero.

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(b) for every x ∈ V , there must exists an n0 ∈ N such that xn0

=‖ x ‖1.(c) for every x ∈ V , there must exists an n0 ∈ N such that xn0

=‖ x ‖2.(d) the dimension of V is uncountable infinite.

7. Let A,B ⊆ R be nonempty sets. Which of the following is wrong?

(a) P (A) ∩ P (B) = P (A ∩B).

(b) P (A) ∪ P (B) ⊆ P (A ∪B).

(c) P (A) ∪ P (B) = P (A ∪B).

(d) P (A ∩B) ⊆ P (A) ∩ P (B).

8. Let us define the density ofA ⊆ N with respect toN equal to limn→∞

|{x∈A:x≤n}|n

.

Then

(a) the density of the set {2, 4, 6, ...} is 2.

(b) there exists a finite subset of N with positive density.

(c) if the density of a set is zero, then it must be a finite set.

(d) for each 0 ≤ α ≤ 1, there exists a set with density equal to α.

9. Let A :={

mn1+m+n

: m,n ∈ N}

. Then

(a) supA < 1 and inf A = 1/3.

(b) supA ∈ A and inf A /∈ A.

(c) supA = 1 and inf A = 0.

(d) supA does not exist and inf A ∈ A.

10. Let A be the collection of all circles in the complex plane having rationalradii and centers with rational coordinates. Let B be the collection ofdisjoint intervals of positive length. Then

(a) A is countable but B is uncountable.

(b) B is countable but A is uncountable.

(c) both A and B are countable.

(d) both A and B are uncountable.

11. Let f : X → Y be a function. Pick the odd statement.

(a) f(A ∩B) = f(A) ∩ f(B), for all A,B ∈ X.

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(b) f−1(f(A)) = A, for all A ⊆ X.

(c) there exists a, b ∈ X such that a 6= b, f(a) = f(b).

(d) for all disjoint A,B ⊆ X, f(A) ∩ f(B) = ∅.

12. The range of the function f : R→ R, defined by f(x) = x−[x]1+x−[x] , where

[.] denotes the greatest integer function,

(a) has its supremum equal to 1.

(b) does not have its minimum element.

(c) is unbounded below.

(d) does not have its maximum element.

13. Pick the series which converge.

(a)∑

sin 1n.

(b)∑n2.

(c)∑

1n2 .

(d)∑

1n.

14. If |A| = m, |B| = 2, then the number of onto functions from A to B is

(a) 2m.

(b) 2m − 1.

(c) 2m − 2.

(d) 2m − 4.

2 Multi-correct Questions

15. Consider a sequence (an), where an = (1+√5

2)n. Define a sequence (bn)

by bn = d(an,Z) where d(an,Z) = inf{d(an, x) : x ∈ Z}. Then

(a) the sequence (an) is not convergent.

(b) the sequence (bn) is convergent.

(c) the sequence (bn) is bounded.

(d) the sequence ( 1an

) is convergent.

16. Let (an) be a sequence given by an = nα sinn, where α ∈ R. Then

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(a) (an) has a convergent subsequence, for α = 1/2.

(b) (an) does not have a bounded subsequence, for α = 3/2.

(c) (an) has a convergent subsequence, for all α < 0.

(d) for α = 0, and c ∈ (−12, 12), there exists a subsequence (ank

) of(an), such that ank

→ c.

17. Let∑an be a series with real-valued terms. Define

pn :=|an|+ an

2, qn :=

|an| − an2

, n ∈ N.

Which of the following hold(s) true?

(a) If∑an is conditionally convergent, then both

∑pn and

∑qn

diverge.

(b) If∑an is conditionally convergent, then one of

∑pn and

∑qn

may converge.

(c) If∑|an| is convergent, then both

∑pn and

∑qn converge.

(d) If∑|an| is convergent, then one of

∑pn and

∑qn may diverge.

18. Let (an) be a bounded sequence in R. Suppose S denotes the range of(an). Then

(a) there exist convergent and monotone subsequences (ank) and (amk

)such that lim

n→∞|ank− amk

| = supS − inf S.

(b) if (an) is divergent, there exist convergent and monotone subse-quences (ank

) and (amk) such that lim

n→∞|ank− amk

| > 0.

(c) there exist an α ∈ R such that for every m ∈ N and ε > 0 thereexists an n0 > m such that |an0 − α| < ε.

(d) the collection {x ∈ R : there exists a subsequence (ank) of (an)

such that (ank)→ x} is a compact set.

19. Let G be a group of all maps from the closed interval [0, 1] to Z. Thesubgroup H = {f ∈ G : f(0) = 0}

(a) is countable.

(b) is uncountable.

(c) has countable index.

(d) has uncountable index.

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20. Which of the following series is(are) convergent?

(a)∑

[12− 1

πtan−1(n)].

(b)∑ sin(n−

√n2+n)

n.

(c)∑

1+5n

1+6n.

(d)∑ sin

√n√

n.

21. Let a, b ∈ Z. The functions f : N ×N → N and g : Z × Z → Z givenby f(x, y) = 2x−1(2y − 1) and ga,b(x, y) = ax+ by respectively. Then

(a) f is one-to-one.

(b) f is onto.

(c) there exists a, b ∈ Z such that ga,b is onto.

(d) there exists a, b ∈ Z such that ga,b is one-to-one.

22. Which of the following statement(s) is(are) true in relation to theCauchy sequences in R?

(a) There exists a Cauchy sequence with rational terms converging toeπ.

(b) If (an) is Cauchy, and its subsequence converges to α, then (an)→α.

(c) There exists a Cauchy sequence for which there exist two subse-quences, converging to two distinct point.

(d) For each Cauchy sequence, there exists a monotone convergentsubsequence.

23. Consider the series∑an and

∑bn, where an = 1

n(logn)4and bn = 1

n.

Then

(a)∑an is convergent.

(b)∑an has the bounded sequence of partial sums.

(c)∑bn has the bounded sequence of partial sums.

(d)∑bn has a convergent sub-series.

24. Which of the following sets is(are) uncountable?

(a) {√

2x : x ∈ Q}.(b) {x+

√2y : x, y ∈ Z}.

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(c) a nonempty open set in R.

(d) {m2n

: m,n ∈ Z}.

25. Let (an) be a real sequence defined by an :=n∑k=1

( 1k

+ sin kk2− log n). Then

(a) (an) converges.

(b)∞∑n=1

an does not converge.

(c) (an −n∑k=1

sin kk2

) converges to zero.

(d)∞∑n=1

(an −n∑k=1

sin kk2

) does not converge.

26. Which of the following hold(s) for real sequences (an) and (bn)?

(a) If (an) and (anbn) are convergent, then so is (bn).

(b) If (an) and (bn) are unbounded, then so is (an + bn).

(c) If (an) is unbounded, an 6= 0, then ( 1an

) may be bounded.

(d) If (an) is unbounded, then limn→∞

|an| =∞.

Answer Key

Single-Correct Questions1. D 2. C 3. B 4. C 5. B 6. B 7. C 8. D9. D 10. C 11. C 12. D 13. C 14. CMulti-Correct Questions15. A,B,C,D 16. A,C,D 17. A,C 18. A,B,C,D 19. B,C20. C 21. A,B,C 22. A,B,D 23. A,B,D 24. C 25. A,B,D 26. C

Best Wishes from Praveen Chhikara...

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“The sole meaning of life is to serve humanity” - Leo Tolstoy

Time duration: 80 minutesMax. points: 115.50 -Praveen Chhikara

Topics: Continuity, Uniform Continuity, Differentiability

Instructions:





1. Suppose f : R → R is a function and F := {x ∈ R : f(x) > 0}. Thenfind the valid statement.

(a) There exists a continuous function f for which F is infinite andcontains no open interval.

(b) There exists a continuous function f for which F is a countableinfinite set.

(c) There exists a continuous function f for which F is finite withmore than one element.

(d) If f is continuous, and F 6= ∅, then F is uncountable.

2. Which of the following statements implies the continuity of the functionf : R→ R?

(a) f 2 is continuous.

(b) f(f(x)) is continuous.

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(c) ef(x) is continuous.

(d) |f | is continuous.

3. Let f : R→ R be a monotone function. The cardinality the set of{a ∈ R : lim

x→a−f(x) 6= lim

x→a+f(x)} cannot be

(a) equal to zero.

(b) equal to that of the open interval (0, 1).

(c) equal to that of Z[√

2].

(d) equal to that of N×N.

4. Suppose y ∈ R. Let fy : R→ R be a function defined byfy(x) = d(x, y), where d(x, y) = |x− y|. Then

(a) fy is not continuous for some y ∈ R.

(b) fy is uniformly continuous for all y ∈ R.

(c) there exists a y ∈ R such that fy is differentiable.

(d) f ′y is a bounded function for all y ∈ R.

5. Which of the following statements is true?

(a) A continuous function f : R → R that is one-to-one must beunbounded.

(b) A continuous function f : R → R that is one-to-one must bedifferentiable.

(c) A continuous function f : R+ → R that is onto must satisfylimn→∞

|f(x)| =∞.

(d) A continuous function f : R → R that is one-to-one must bestrictly monotone.

6. Suppose f : (0,∞)→ R is differentiable and |f ′(x)| < 1. Then

(a) (an), where an = f(23)n, is not convergent.

(b) (bn), where bn = f( nn+1

), is not convergent.

(c) (cn), where cn = f( 1n), is convergent.

(d) there exists an x0 > 0, such that f(x0) = x0.

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7. Suppose A := {x ∈ [0, 1] : x = m2n

for some m ∈ Z, n ∈ N} and

B := Z[√

2] ∩ [0, 1]. Let f : [0, 1]→ R be a function defined by

f(x) =

x+ 3

2; for x ∈ A

2x+ 1; for x ∈ B2; otherwise.

Then

(a) f is continuous at all points x /∈ A ∪B.

(b) f is discontinuous at each point of its domain.

(c) f is differentiable at exactly one point of its domain.

(d) f is continuous at exactly one point but differentiable nowhere.

8. Suppose f : (a, b) → R is differentiable and |f ′(x)| 6= 0 for x ∈ (a, b).Then

(a) f need not be one-to-one.

(b) f is strictly monotone.

(c) f ′ is bounded.

(d) there exists some c, d ∈ (a, b) such that f ′(c)f ′(d) < 0.

9. Suppose f : [0, 1]→ R is a function and S is the range of f . Then

(a) If f is strictly increasing, then S may be a subset of Q.

(b) If f is increasing, then S may be a unbounded.

(c) If f is strictly increasing and continuous, then S may be a subsetof Q.

(d) If f is strictly increasing and continuous, then S cannot be a subsetof R \Q.

10. Let f : R → R be a differentiable function. Suppose f(0) = 1 andf ′(x) ≤ 5 for all x ∈ (0, 7). Then f(2) is atmost equal to

(a) 9

(b) 10

(c) 11

(d) 12

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11. For E ⊂ R, consider the following statements:P : Every continuous function f : E → R is uniform continuous.Q: E is compact.R: Every continuous function f : E → R is bounded.Which of the following is(are) false?

(a) R ; P

(b) P ⇒ Q

(c) Q⇒ P

(d) R⇒ Q

12. The equation x sinx+ cosx− x2 = 0 has

(a) at least two solutions in R.

(b) at most two solutions in R.

(c) exactly one positive real solution.

(d) exactly one negative real solution.

13. Let f : [0,∞) → R be a continuous function. Suppose that f isdifferentiable for all x > 0 and that lim

x→∞f ′(x) = 0. Further suppose

limx→∞

(f(x) + f ′(x)) exists finitely. Then

(a) limx→0+

f( 1x) exists.

(b) f must be a uniform continuous function.

(c) limx→0+

f ′(x) exists.

(d) f must be a bounded function.

14. For A ⊆ R, the function f : A→ R, f(x) = x2 is a uniform continuousfunction if

(a) A is bounded.

(b) A is unbounded.

(c) A is compact.

(d) A is the set Z of all integers.

15. Let T : Rn → Rm be a linear transformation. Then

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(a) T need not be continuous.

(b) T is differentiable.

(c) T is differentiable but not continuously differentiable.

(d) the partial derivatives of T need not exist.

16. Which of the following is(are) correct inequality(ies)?

(a) ex ≥ ex ∀ x ∈ R.

(b) x1+x≤ log(1 + x) ∀ x > 0.

(c) ex ≥ 1 + x ∀x ∈ R(d) y−x

y≤ log y

xfor 0 < x < y.

17. Let f : R→ R be a continuous and periodic function. Then

(a) f attains its supremum.

(b) f need not be bounded.

(c) there do not exist sequences (xn) and (yn) in R such that|xn − yn| → 0 but |f(xn)− f(yn)|9 0.

(d) if (xn) is a Cauchy sequence in R, then (f(xn)) is also a Cauchysequence.

18. Let f : [0, 1]→ R be a bounded function. Which of the following maybe true statement(s)?

(a) f attains infimum but not supremum.

(b) f attains supremum but not infimum.

(c) f attains neither an infimum nor a supremum.

(d) f attains its supremum and infimum on any nonempty open in-terval A ⊂ [0, 1].

19. Let f : R → R be the characteristic function of the Cantor set C.Then,

(a) limx→a

f(x) = 1, if a ∈ C.

(b) limx→a

f(x) = 0, if a /∈ C.

(c) f is continuous at a if a ∈ C.

(d) f is continuous at a if a /∈ C.

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20. A function f : R→ R is said to be symmetrically continuous at x ∈ Rif limh→0

[f(x+ h)− f(x− h)] = 0. Which of the following is(are) true?

(a) There exists a continuous function at x which is not symmetricallycontinuous at x.

(b) A symmetrically continuous function at x must be continuous atx.

(c) If f is symmetrically continuous at x, then limh→0

f(x+h)−f(x−h)h

= 0.

(d) A differentiable function must be symmetrically continuous at allx ∈ R.

21. Suppose IVP refers to the Intermediate Value Property of functions.Let f : (a, b)→ R be a function. SupposeA = {y ∈ R : card({x : f(x) =y}) =∞}. Which of the following is(are) true?

(a) if f is strictly monotone, then A must be empty.

(b) if f has IVP on (a, b) and is discontinuous, then A must benonempty.

(c) if f has IVP, then A can be non-compact.

(d) if f has IVP, then A can be compact.

22. Let f : R→ R be a continuous function. Then

(a) if f is uniformly continuous on both (−∞, 0] and [0,∞), then fis uniformly continuous on R.

(b) if f is uniformly continuous on both (−∞, 0] and [1,∞), then fis uniformly continuous on (−∞, 0] ∪ [1,∞).

(c) if f is uniformly continuous on each of the compact setsX1, X2, . . . , Xn,

then f is uniformly continuous onn⋃i=1

Xi.

(d) if f is uniformly continuous on each of the compact intervals

X1, X2, . . . , Xn, . . ., then f is uniformly continuous on∞⋃i=1

Xi.

23. Which of the following is(are) true about real-valued functions?

(a) There exists a strictly monotone function f on [0, 1] that is dis-continuous at each point of (0, 1)∩Q and continuous at each pointof (0, 1) ∩ (R \Q).

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(b) If f is continuous on an open interval and has no local maximumor local minimum, then f must be monotone.

(c) There exists a monotone function on [0, 1] that is discontinuousonly at points of (0, 1) ∩ (R \Q).

(d) If f : [a, b] is monotone, then f must be a continuous on an un-countable dense set of [0, 1].

24. Let f : I → R be a function where I be a nondegenerate interval in R.We define the oscillation of f on I by ωf(I) = sup

x,y∈I|f(x) − f(y)| and

oscillation of f at x0 ∈ I by ωf (x0) = infδ>0

ωf((x0 − δ, x0 + δ)).

(a) f is continuous at x0 ∈ I if, and only if ωf (x0) = 0.

(b) ωf(I) = supx∈I

f(x)− infx∈I

f(x).

(c) ωf (x0) = limx→x0

sup f(x)− limx→x0

inf f(x).

(d) ωf (x0) = limδ→0+

ωf((x0 − δ, x0 + δ)).

25. Pick the correct statement(s).

(a) There exists a function f : R→ R such that

f ′(x) =

{0; if x < 0

1; if x > 0.

(b) If f : R → R is differentiable and there is an M > 0 such that|f ′(x)| < M ∀ x ∈ R, then there exists an ε > 0 for which x+εf(x)is a one-to-one function.

(c) If f : (a, b)→ R is differentiable, then f ′(x) = limh→0

f(x+h)−f(x−h)2h

.

(d) If f : (a, b) → R is a function such that limh→0

f(x+h)−f(x−h)2h

exists

for all x ∈ (a, b), then f is a differentiable function.

26. Let f : R→ R be a differentiable function. If f ′(x) 6= 1 for all x ∈ R,then

(a) f has at least one fixed point.

(b) f may have more than one fixed point.

(c) f has a unique fixed point.

(d) f may have no fixed point.

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27. Let f : [a, b]→ R be a differentiable function. Suppose there does notexist any x ∈ [a, b] for which f(x) = f ′(x) = 0.Let A := {x ∈ [a, b] : f(x) = 0} be nonempty. Then A

(a) is closed.

(b) is compact.

(c) is necessarily finite.

(d) may be open.

28. Suppose C denotes the Cantor set. Consider the function f : [0, 1]→ R

defined by

f(x) =

{1; if x ∈ C,0; if x /∈ C.

Then

(a) if a ∈ C, limx→a

f(x) exists.

(b) f is continuous at uncountable number of points.

(c) f is discontinuous at most countable infinite number of points.

(d) f is differentiable no-where.

Answer Key

Single-Correct Questions1. D 2. C 3. B 4. B 5. D 6. C 7. D 8. B9. D 10. CMulti-Correct Questions11. A,B. 12. A,B,C,D 13. A,B,D 14. A,B,C,D 15. B16. A,B,C,D 17. A,C,D 18. A,B,C,D 19. B,D 20. D21. A,C,D 22. A,B,C 23. A,B 24. A,B,D 25. B,C 26.A,C 27. A,B,C 28. B,D


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“Every great dream begins with a dreamer. Always remember, you havewithin you the strength, the patience, and the passion to reach for the starsto change the world.” - Harriet Tubman


Topics: Uniform Convergence, Riemann Integration

Instructions:





1. Let (fn) be a sequence of functions on [0, 1] defined by

fn(x) =

{n; if 0 < x < 1

n

0; if x = 0 or x ≥ 1n.

Suppose (fn)→ f. Then

(a) limn→∞

1∫0

fn(x)dx = 0.

(b)1∫0

f(x)dx 6= 0.

(c) (fn)→ f uniformly on [0, 1].

(d)1∫0

fn converges uniformly on [0, 1].

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2. Suppose f : [−1, 1]→ R is a function given by

f(x) =

{1; if x 6= 1,

0; if x = 1.

If g is the indefinite integral of f , then

(a) g is not of bounded variation on [−1, 1].

(b) g is differentiable at x = 1.

(c) g is not Riemann integrable on [0, 1].

(d) g is not continuous on [0, 1].

3. Suppose for n ∈ N,

fn(x) =

0; if x ≤ 0,

nx; if 0 ≤ x ≤ 1n,

1; if x ≥ 1n.

The sequence (fn) of functions

(a) converges to the zero function on [0, 1].

(b) converges uniformly on [0, 1].

(c) converges to a function which is discontinuous at x = 1.

(d) converges to a non-differentiable function on [0, 1].

4. For the series∞∑n=1

cos(2nx)2n

for all x ∈ R, which of the following is false?

(a) The series converges uniformly on R.

(b) The sum of the series is continuous on R.

(c) The sum of the series is monotone in some nonempty interval ofR.

(d) The sum of the series is not differentiable at any point of R.

5. Suppose that f : [a, b] → R is Riemann integrable. Let F (x) =x∫a

f(t)dt. Which of the following is correct?

(a) If f is discontinuous at x0 ∈ [a, b], then F (x) is discontinuous atx0.

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(b) Even if f is discontinuous at x0 ∈ [a, b], F (x) is continuous at x0but is not differentiable at x0.

(c) The function F (x) is continuous but not necessarily a Lipschitzfunction on [a, b].

(d) F ′(x), if it exists, need not be equal to f(x).

6. Let {α1, α2, . . .} be an enumeration of the set Q of all rational numbers.If (fn) is a sequence of functions given by

fn(x) =

{1, if x ∈ {α1, . . . , αn}0, otherwise.

Then

(a) (fn) converges uniformly on [0, 1].

(b) (fn) converges to a Riemann integrable function on [0, 1].

(c) f(x) = limn→∞

limm→∞

cos2m(n!πx) for all x ∈ R.

(d) f(x) = limn→∞

fn(x) is a function of bounded variation on [0, 1].

7. Let f : [0, 1] → R be a bounded function. Suppose ℘[0, 1] denotesthe collection of all partitions of [0, 1]. Further suppose U(f, P ) andL(f, P ) denote respectively the upper sum and the lower sum of f forP ∈ ℘[0, 1]. If there exist sequences (an) in R and (Pn) in ℘[0, 1] suchthat

U(f, Pn)− L(f, Pn) < an,

then which of the following guarantees the Riemann integrability of fon [0, 1]?

(a) an = n2n+1

.

(b) an = nlogn

.

(c) an = nn+1

.

(d) an, where∑|an| <∞.

8. Consider the series

f(x) =∞∑n=0

xn

non (−1, 1).

Then

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(a) f is not continuous at all x ∈ (−1, 1).

(b) f is continuous but need not be differentiable at all x ∈ (−1, 1).

(c) f ′(x) =∞∑n=1

xn−1 for all x ∈ [−α, α] where, α is any element of

(0, 1).

(d) the given series is not uniformly convergent on [−α, α] for some α ∈(0, 1).


9. Let f : [0, 1] → R be a continuous function. Suppose1∫0

g(x)f(x)dx =

0 for each continuous function g : [0, 1]→ R. Then

(a) f must be differentiable everywhere.

(b) f need not be Riemann integrable on [0, 1].

(c) f must be Riemann integrable but need not be of bounded varia-tion on [0, 1].

(d) f ′ must be continuous.

10. Suppose (fn) is a sequence of functions on R given by fn(x) = x1+nx2 for all x ∈

R. Let f(x) = limn→∞

fn(x) and g(x) = limn→∞

f ′n(x) for all x ∈ R. Then

(a) (fn) converges uniformly on R.

(b) (f ′n) converges uniformly on R.

(c) f ′(0) = g(0).

(d) f ′(a) 6= g(a) for some nonzero real number a.

11. Let fn, n ∈ N and f be differentiable functions with fn → f on [0, 1].Suppose f is bounded and Riemann integrable. Then which of the

following ensure(s) that limn→∞

1∫0

fn =1∫0

f?

(a)1∫0

fn converges uniformly on [0, 1].

(b) fn → f uniformly on [0, 1].

(c) (fn) does not converge uniformly on [0, 1] but it converges uni-formly on [a, 1], for all a ∈ (0, 1).

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(d) (f ′n) converges uniformly on [0, 1].

12. Which of the following function is(are) Riemann integrable on [0, 1]?

(a) f(x) = e−x.

(b) g(x) = sin x.

(c) h(x) = χQ.

(d) k(x) = |x|.

13. Which of the following is(are) uniformly convergent series?

(a)∞∑n=1

rn cosnt, where r ∈ (−1, 1) on R.

(b)∞∑n=1

xn(1+nx2)

on R.

(c)∞∑n=1

sinnx√n

on R.

(d)∞∑n=1

xn

n!on [−a, a] where a ∈ (0, 1).

14. Suppose f : [0, 1] → (0,∞) is a continuous function. Then which ofthe following is(are) false?

(a) f is of bounded variation on [0, 1] if, and only if f has boundedderivative.

(b) f may be of bounded variation on [0, 1] if f is not Riemann inte-grable on [0, 1].

(c) f is of bounded variation on [0, 1] if f is continuous on [0, 1].

(d) 1/f is of bounded variation on [0, 1] if f is of bounded variationon [0, 1].

15. A function f : [a, b]→ R is said to be absolutely continuous on [a, b] iffor every ε > 0, there exists a δ > 0 such that

n∑k=1

|f(bk)− f(ak)| < ε

for every n disjoint open subintervals (ak, bk) of [a, b], n ∈ N andn∑

k=1

(bk − ak) < δ.

Choose the correct statement(s).

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(a) There exists an absolutely continuous function on [a, b] which isnot of bounded variation on it.

(b) There exists a continuous function on [a, b] which is of boundedvariation but not absolutely continuous on it.

(c) There exists a Lipschitz continuous function on [a, b] which is notabsolutely continuous on it.

(d) There exists an absolutely continuous function on [a, b] which isnot continuous on it.

16. Consider two sequences (fn) and (gn) of functions respectively given by

fn(x) = x(1 +1

n) if x ∈ R, and

gn(x) =

{1n; if x ∈ R \Q or x = 0,

b+ 1n; if x = a

b, where a, b ∈ Z, b > 0 and gcd(a, b) = 1.

Suppose hn(x) = fn(x)gn(x). Then

(a) (fn) converges uniformly on every bounded interval.

(b) (gn) converges uniformly on every bounded interval.

(c) (hn) converges uniformly on every bounded interval.

(d) (hn) converges to a function which is continuous no-where.

17. Let f : [0, 1]→ (0,∞) be continuous. Suppose sup{f(x) : x ∈ [0, 1]} =M . Then

gn(x) := limn→∞

( 1∫0

[f(x)]ndx

)1/n

(a) is convergent to the zero function.

(b) is uniformly convergent.

(c)∑gn(x) is convergent.

(d)∑ gn(x)

2nis uniformly convergent.

18. Suppose fn : R→ R, n ∈ N are continuous functions. Let (fn) � f onR. Then

(a) limn→∞

∞∫0

fn =∞∫0

f.

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(b) limn→∞

1∫0

fn =1∫0

f.

(c) limn→∞

1− 1n∫

0

fn =1∫0

f.

(d) if each fn are differentiable, then f is differentiable.

19. Suppose (fn) is a sequence of functions on A ⊂ R, converging uniformlyto a function f : A→ R. Then

(a) if each fn is bounded on A, then f is bounded on A.

(b) if each fn is bounded on A, then (fn) is uniformly bounded on A.

(c) if f is bounded on A, then (fn)∞n=k is uniformly bounded on A forsome k ∈ N.

(d) if f is bounded on A, then all but finitely many terms of (fn) arebounded on A.

20. For a function f : [0, 1] → R, suppose f(0) > 0, and f(x) ≤ f(y)for x ≤ y. Further suppose f has no fixed point. Let A = {x ∈[0, 1] : f(x) > x}. Then

(a) supA ∈ A.(b) f is Riemann integrable on [0, 1].

(c) f(1) must be greater than 1.

(d) f is of bounded variation on [0, 1].

Answer Key

Single-Correct Questions1. D 2. B 3. C 4. C 5. D 6. C 7. D 8. CMulti-Correct Questions9. A,D 10. A 11. B,C,D 12. A,B,D 13. A,B,C,D 14.A,B,C 15. B 16. A,B,C,D 17. B,D 18. B 19. A,B,C,D20. A,B,C,D


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“To be successful in life what you need is education, not literacy and degrees.”- Munsi Premchand


Topics: Metric Spaces, Functions of Several Variables

Instructions:





1. The cardinality of the collection of all continuous functions from R toR is equal to that of

(a) Q.

(b) R.

(c) P(R).

(d) P(P(R)).

2. Let E be a nonempty set in a discrete metric space. Then E is singletonif and only if

(a) E is closed.

(b) every continuous function f : E → R is constant.

(c) E is connected.

(d) E is open.

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3. Which of the following is compact in the collection of all n×n matricesover R?

(a) GL(n,R), the collection of all n× n invertible matrices over R.(b) SL(n,R), the collection of all n×n matrices, with determinant 1,

over R.

(c) O(n,R), the collection of all n× n orthogonal matrices over R.(d) the collection of all n× n nilpotent matrices over R.

4. The direction in which the function defined by f(x, y) = xe2y−x in-creases most rapidly, at the point (2, 1) is

(a) i− 4j.

(b) −i+ 4j.

(c) −i− 4j.

(d) i+ 4j.

5. If the function

f(x, y) =

{x3+y3

x2+y2; if (x, y) 6= (0, 0),

A; if (x, y) = (0, 0).

is continuous at the origin, then A

(a) equals 0.

(b) equals 1.

(c) equals −1.

(d) does not exist.

6. Let A and B be two sets defined byA := {f : R→ R | f is continuous, f(x+y) = f(x)+f(y) ∀ x, y ∈ R},B := {f : R → R | f is one-to-one, f(x) ∈ Q, if x /∈ Q and f(x) /∈Q, if x ∈ Q}.Then

(a) A is countable and B is uncountable.

(b) A is uncountable and B is countable.

(c) both A and B are countable.

(d) neither A nor B is countable.

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7. Let H be a subgroup of the group G = C∗ of all nonzero complexnumbers. Suppose H is open, when considered as a set in the metricspace C. The order of the quotient group G/H is

(a) 1.

(b) 2.

(c) countable infinite.

(d) uncountable.

8. Which of the following is not connected in R?

(a) {x ∈ R : sin2 x+ cos2 x = 1}.(b) {x ∈ (0,∞) : x+ 1

x≥ 2}.

(c) {x ∈ R : sinx > 0}.(d) {x ∈ R : x2 − 3x+ 2 < 0}.

9. The collection of all limit points of the setA = {(sinm, cosn) ∈ R2 : m,n ∈ N} is

(a) the empty set ∅.(b) [0, 1]× [0, 1].

(c) [−1, 1]× [−1, 1].

(d) A.

10. Suppose T : R2 → R3 is a function given by

T (x, y) = (x+ 2y, sinx, ey).

The derivative DT (x, y) : R2 → R3 is

(a) [1 cos x 02 0 ey

](b) [

1 cos x 02 0 −ey

](c) 1 2

cosx 00 ey

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(d) 1 2cosx 0

0 −ey

11. Which of the following is a compact set in R3?

(a) {(x, y, z) ∈ R3 : x2 + y2 = 1}.(b) {(x, y, z) ∈ R3 : x2 + y2 = 2z}.(c) {(t cos t, t sin t, t) ∈ R3 : t ∈ R}.(d) {(x, y, z) ∈ R3 : − 1 ≤ x, y ≤ 1, z = 0}.


12. Let (X, d) be a metric space. Suppose B(a, ε) := {x ∈ X : d(x, a) <ε} and B[a, ε] := {x ∈ X : d(x, a) ≤ ε} are respectively the open balland the closed ball in X, centered at a both with radius ε > 0. SupposeA denotes the closure of A in X. Then

(a) B(a, ε) = B[a, ε], where ε > 0, a ∈ X.(b) if a ∈ X, 0 < ε1 < ε2, then B(a, ε1) $ B(a, ε1).

(c) it may be true that B(a, 12) = B(a, 3

4).

(d) it may be true that B(a, ε) = B[a, ε], for some ε > 0.

13. Let X be a nonempty set. Suppose d : X ×X → R is a metric. Then

(a) if d(xn, yn)→ d(x, y), then (xn, yn)→ (x, y).

(b) if (xn, yn)→ (x, y), then d(xn, yn)→ d(x, y).

(c) if G is open in X ×X, then d(G) is open in R.

(d) if U is open in R, then d−1(U) is open in X ×X.

14. Suppose C denotes the set of all complex numbers. The cardinality ofa discrete set in the metric space C

(a) can be 1.

(b) can be finite and more than 1.

(c) can be countable infinite.

(d) cannot be uncountable.

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15. For a set A in a metric space, which of the following is(are) correct?

(a) If A is bounded, then A is totally bounded.

(b) If A is totally bounded, then A is totally bounded.

(c) If A is totally bounded, then A is totally bounded.

(d) If A ⊆ Rn is bounded, then A is totally bounded.

16. Let X be a metric space equipped with a metric d. Then,

(a) for a fixed point x0 ∈ X, the function f(x) = d(x0, x) is uniformlycontinuous.

(b) for a fixed subset A ⊆ X, the function f(x) = d(A, x) is uniformlycontinuous.

(c) for an arbitrary open set A ⊆ X, there exists a continuous functionf : X → R which vanishes exactly for points of A.

(d) for an arbitrary closed set A ⊆ X, there exists a continuous func-tion f : X → R which vanishes exactly for points of A.

17. Let Y be the subspace of l∞ with Y consisting of all sequences withterms zeros and ones only. Then

(a) Y is compact.

(b) Y is complete.

(c) Y is totally bounded.

(d) any sequence in Y is eventually constant.

18. Which of the following is(are) metric(s)?

(a) d on C[a, b] given by d(x, y) =b∫a

|x(t)− y(t)|dt.

(b) d on {x = (x1, x2, x3) : xi ∈ {0, 1}} where d(x, y) is equal to thenumber of tuples, where x and y differs.

(c) d on R given by d(x, y) = | x2

x2+1− y2

y2+1|.

(d) d on the collection of all real sequences x = (xn), given by

d(x, y) =

{0; if x = y

1min{k:xk 6=yk}

; if x 6= y.

19. Which of the following is(are) not complete metric space(s)?

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(a) Y , a subspace of l∞, where Y contains sequences with at mostfinitely many nonzero terms.

(b) R, with metric d(x, y) = | arctanx− arctan y|.(c) X = (0,∞), with metric d(x, y) = |m−1 − n−1|.(d) X, the collection of all continuous function from [0, 1] to R with

metric d(f, g) =1∫0

|f(x)− g(x)|dx.

20. Suppose A = {(x, y) ∈ R2 : x ∈ [0, 1], y = xn

for n ∈ N} and B ={(x, y) ∈ R2 : x ∈ [1/2, 1], y = 0}. Then

(a) A is not connected.

(b) A ∪B is path-connected.

(c) A ∪B is connected.

(d) A ∪B is path-connected.

21. Let X be a compact metric space. Suppose for given a, b ∈ X, thereexist elements x1, x2, . . . , xn ∈ X, where x1 = a and xn = b such thatd(xi, xi+1) < ε.

(a) X is totally bounded.

(b) X is connected.

(c) X is path-connected.

(d) X is separable.

22. Which function(s) f : R2 → R of the following is(are) uniformly con-tinuous?

(a) f(x, y) = x+ y.

(b) f(x, y) = x− y.

(c) f(x, y) = xy.

(d) f(x, y) = min{x, y}.

23. Let X be metric space, Y a complete metric space. Suppose C(X, Y )denotes the space of bounded continuous functions from X to Y , withmetric d(f, g) = sup

x∈Xd(f(x), g(x)). Then

(a) if (fn) is Cauchy in C(X, Y ), then (fn) converges uniformly.

(b) C(X, Y ) is complete.

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(c) C(X, Y ) is totally bounded.

(d) C(X, Y ) is compact.

24. Consider the Lp, p > 1 space. Let W be a proper subspace of Lp. Then

(a) W cannot be open.

(b) Lp spaces are connected, for all p.

(c) Lp spaces are path-connected, for all p.

(d) there exists an ε > 0 such that Bε(0) = {x : ‖x‖ < ε} ⊆ W .

25. Suppose A ⊆ Rn is nonempty and not closed. Then

(a) there exists a sequence (xn) in A that converges to a point x0 /∈ A.

(b) there is a continuous function f : A→ R which is unbounded.

(c) there exists an x0 ∈ Rn such that, for each n ∈ N, there is anxn ∈ A satisfying ‖ x0 − xn ‖≤ 1√

n.

(d) there exists an x0 ∈ Rn, and a d > 0 such that ‖ x− x0 ‖>√d for

all x ∈ A.

26. Which of the following is(are) not locally compact space(s)?

(a) R3

(b) {(x, y) ∈ R2 : xy > 0} ∪ {(0, 0)}.(c) {(x, y) ∈ R2 : x2 + y2 < 1}.(d) R \Q.

27. Suppose X = [0, 1]. Then

(a) ( 1n)→ 1

2in some metric space (X, d).

(b) ( 1n)→ 3

4in some metric space (X, d).

(c) ( 1n)→ 1 in some metric space (X, d).

(d) ( 1n) is not Cauchy in some metric space (X, d).

Answer Key

Single-Correct Questions1. B 2. C 3. C 4. B 5. A 6. B 7. A 8. C 9. C10. C 11. DMulti-Correct Questions

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12. C,D 13. B,D 14. A,B,C,D 15. B,C,D 16. A,B,D 17.B,D 18. A,B,D 19. A,B,C,D 20. C,D 21. A,B,D 22. A,B,D23. A,B 24. A,B,C 25. A,B,C 26. B,D 27. A,B,C,D


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“Our knowledge should work for humanity rather than for thepeople involved in disastrous works.”: Praveen Chhikara

Time duration: 120 minutesMax. points: 178.75 -Praveen Chhikara

Topics: Set theory, Countable sets, Sequences and Series, Point-set topol-ogy, Continuity, Uniform Continuity, Differentiability, Uniform Convergence,Riemann Integration, Metric Spaces.

Instructions:

1. There are two sections, first of “Multi-correct questions” and second of“Single-correct questions”.

2. Each question of the first section has one or more correct option(s).One question carries 4.75 marks, and there is no negative marking inthis section.

3. Each question of the second section has one correct option. One ques-tion carries 3 marks, and a wrong choice leads to NEGATIVE 0.75marks.


Multi-correct Questions (Question 1-25)

1. Let f : R→ R be a function. Consider the set

S = {a ∈ R : limx→af(x) exists and limx→af(x) 6= f(a)}.

Then, the set S(a) may be countable infinite.(b) may be uncountable infinite.(c) is at-most countable infinite.(d) is always infinite.

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2. Let A and B be subsets of a metric space X. Then which of the fol-lowing is(are) true?(a) (A ∩B)o = Ao ∩Bo.(b) A ∩B = A ∩B.(c) Ao ∪Bo ⊆ (A ∪B)o.(d) If X \B is closed, then A ∩B ⊆ A ∩B.

3. Let f : X → Y be a continuous function. Then(a) f−1(B) is open in X for each open set B in Y .(b) f−1(B) is closed in X for each closed set B in Y .(c) f(A) may not be open in Y for each open set A in X.(d) f−1(Bo) ⊆ (f−1(B))o.

4. Consider a function f : R→ R. Which of the following is(are) true?(a) If f is bijective continuous, then f−1 is also continuous.(b) If f(f(x)) is continuous and strictly decreasing, then f is continuous.(c) If f is bijective continuous and f(I) is an interval for each interval I ⊆ R,then f−1 is also continuous.(d) If f is strictly increasing and continuous, then f−1 is also continuous.

5. The function f : R→ R defined by

f(x) =

{x2 − 1, x ∈ Q

3(x− 1), otherwise

(a) is no-where continuous.(b) is continuous at x = 1, 2.(c) is differentiable at x = 1.(d) is differentiable at x = 2.

6. The function f : R→ R defined by

f(x) =

{x2

+ x2 sin 1x, x 6= 0

0, x = 0

(a) is differentiable everywhere.(b) f is strictly increasing in some neighborhood of zero.(c) f

′(0) = 1

2.

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(d) f is not differentiable at zero.

7. Pick the valid statement(s) about the nature of real sequences.(a) There exist sequences (xn) and (yn) which both diverge, but (xn + yn)converges.(b) There exists a sequence (xn), where xn 6= 0, converges, but ( 1

xn) diverges.

(c) There exist sequences (xn) and (yn) such that (xn) converges, (yn) di-verges, and (xn + yn) converges.(d) There exist sequences (xn) and (yn) such that (xn) and (xnyn) both con-verge, but (yn) diverges.

8. For a sequence (an), where an =∑n

k=1(−1)k2k

,(a) lim sup an = 3.(b) lim inf an = −1

3.

(c) lim sup an = −13.

(d) lim inf an = −3.

9. Consider the set S = {(x, sin 1x) : x > 0} ∪ {(0, 0)}. Then

(a) S is not path connected.(b) S is not path connected but connected.(c) S is connected.(d) S ∪ {(0, y) : 0 < y < 1

2} is connected.

10. Which of the following statements is(are) correct?.(a) A Riemann integrable function on [a, b] must be of bounded variation.(b) A function with finite arc-length over [a, b] must be Riemann integrable.(c) A continuous function on [a, b] must of bounded variation.(d) A polynomial function on [a, b] must of bounded variation.

11. The integral∫∞1

sinx√xdx

(a) converges.(b) converges absolutely.(c) converges but not absolutely.(d) diverges.

12. Which of the following statements about sets in M(n,R) is(are) true?(a) The set O(n) of all n× n orthogonal matrices over R is compact.(b) The set GL(n,R) of all n×n matrices over R with determinant nonzero

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is open.(c) The set SL(n,R) of all n×n matrices over R with determinant equal toone is open(d) The set of all n× n nilpotent matrices over R over R is closed.

13. Which of the following is(are) TRUE?(a) {(x, y, z) ∈ R3 : x3 − y3 = 0} is connected but not compact.(b) {(x, y, z) ∈ C3 : x2 + y2 + z2 = 0} is compact.(c) {(x, y) ∈ R2 : sinx = 1} ∪ {(x, 2) : x ∈ R} is connected.(d) {(x,

√x sin 1

x) : x > 0} is connected.

14. Suppose x0 ∈ R. Pick out the true statement(s).(a) There does not exist a differentiable function f : R → R such thatf′(x0) 6= lim x→x0f

′(x).

(b) There does not exist a continuous function f : R → R such thatf(x0) 6= lim x→x0f(x).(c) If f : R → R is a function, and lim x→x0f

′(x) exists and is finite, then

f′(x0) exists but f

′is not continuous at x0.

(d) If f : R → R is a function, and lim x→x0f′(x) exists and is finite, then

f′(x0) exists and f

′is continuous at x0.

15. Consider a function f : R→ R. Then(a) if f is monotone, then one-sided limit exists at each point.(b) if f is continuous and I ⊆ R is an interval, then f(I) is also an intervalor a singleton.(c) if f is differentiable, then f holds the intermediate value property.(d) if f is differentiable and monotone, then f

′must be continuous.

16. Let (fn) be a sequence of functions converging to f on [a, b]. Then(a) if each fn is increasing on [a, b], then so is f .(b) if each fn is bounded on [a, b], then so is f .(c) if each fn is everywhere discontinuous on [a, b], then so is f .(d) if each fn is continuous on [a, b], then so is f .

17. Suppose (fn) is a sequence of functions. Then(a) if fn → f pointwisely on a finite set S, then the convergence is uniformon S.(b) if fn → f uniformly on sets S1 and S2, then the convergence is uniform

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on S1 ∪ S2.(c) if fn → f uniformly on each closed interval [a, b] contained in an openinterval (c, d), then the convergence is uniform on (c, d).(d) if (fn) and (gn) both converge uniformly on a set S, then so is (fngn).

18. Which of the following is(are) TRUE in relation to real sequences?(a) The sequence (xn), where xn = (n

2+5n−7n2+1

) sin(2n) has a convergent subse-quence.(b) For any sequence (xn) in (0,∞), the series

∑(xn + 1

xn) diverges to ∞.

(c) A sequence (xn) is bounded if and only if every subsequence of (xn) hasa convergence subsequence.(d) If (xn) is an enumeration of all rational numbers in [0, 1], then (xn) is notconvergent.

19. The equation x4 + 5x3 − 7 = 0 has(a) no real roots.(b) exactly one real root.(c) has at least two real roots.(d) has at most two nonreal roots.

20. Which of the following is(are) true?(a) If f : R → R is a function such that | f ′(x) |≤ 3, then f is uniformcontinuous.(b) There exists a uniform continuous polynomial function f : R → R ofdegree greater than one.(c) A continuous periodic function on R must be uniform continuous.(d) If f is uniform continuous on (a, b), and also on (c, d), then it is uniformcontinuous on (a, b) ∪ (c, d).

21. Which of the following is(are) true?(a) The characteristic function of Q is nowhere continuous.(b) There exists a continuous function f : R → R such that f(x) ∈ Qc forx ∈ Q and f(x) ∈ Q for x ∈ Qc.(c) If two continuous functions f, g : R → R such that f(x) = g(x) for{m2n

: m ∈ Z, n ∈ N}, then f(x) = g(x).(d) There exists a function f : R → R which is discontinuous at exactlypoints of (0, 1).

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22. Tick the valid statement(s).(a) If

∑an converges, so is

∑a4n.

(b) If∑an converges absolutely, then

∑a2n converges.

(c) If∑an converges absolutely, then so is

∑annp for all p ≥ 0.

(d) If∑| an | converges, so is

∑an.

23. Let (xn) be an enumeration of all rational numbers. Suppose (fn) isa sequence of functions on R, defined by

fn(x) =

{1, x = x1, x2, ..., xn0, otherwise

Then(a) (fn) does not converge pointwise.(b) (fn) converges pointwise to the zero function.(c) (fn) converges uniformly on [0, 1].(d) (fn) does not converge uniformly on any interval [a, b].

24.Which of the following is(are) true?(a) The set N is compact in R.(b) An infinite discrete metric space is not compact.(c) The set Q is dense in R.(d) Any finite set in R is closed.25. Let X be a metric space. Suppose there exists a sequence {An} ofnowhere dense subsets in X such that X = ∪∞n=1An. Then(a) there exists a sequence {On} of open dense sets in X such that ∩∞n=1On

is not dense in X.(b) if {Fn} is a sequence of closed sets in X, such that X = ∪∞n=1Fn, thenthe open set ∪∞n=1(Fn)0 is a dense set in X.(c) X is not a complete space.(d) there exists a Cauchy sequence in X, which does not converge.

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Single-correct Questions (Question 26-40)

26. Let {αn} and {βn} be two strictly increasing sequences of natural num-bers such that {αn} ∪ {βn} = N . Which of the following implies the con-vergence of a sequence {xn}?(a) {xαn} and {xβn} both converge.(b) {xαn} and {xβn} are both Cauchy.(c) There exists a real number a such that {xαn} → a and {xβn} → a.(d) There exists real numbers a, b such that {xαn} → a and {xβn} → b.

27. The derived set of { 1n

: n ∈ N} is(a) {0}.(b) {0, 1}.(c) [0, 1].(d) { 1

n: n ∈ N}.

28. Let A ⊆ R be nonempty. Define a set S byS = {x ∈ A : (x, x+ ε) ∩ A = ∅ for some ε > 0}. The set S(a) is finite.(b) is infinite.(c) is finite or countable infinite.(d) may be uncountable.

29. Which of the following statements is false?(a) Not every open cover of (a, b) has a finite subcover.(b) Every open cover of a subset of R has an at-most countable subcover.(c) There exists an open cover of N which has a finite subcover.(d) Every open cover of { 1

n: n ∈ N} has a finite subcover.

30. The boundary of a set A in R(a) is always nowhere dense.(b) may not be nowhere dense, if A is closed.(c) is nowhere dense, if A is open.(d) may contain a nonempty interval, if A is closed.

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31. Let (X, d) be a metric space. Suppose A,B ⊆ X are nonempty. Wedefine the distance between A and B by

d(A,B) = inf{d(x, y) : x ∈ A, y ∈ B}.

Which of the following is the minimum possible requirement for d(A,B) tobe positive?(a) A and B are nonempty, closed, and disjoint sets.(b) A and B are nonempty, closed, disjoint, and bounded sets.(c) A and B are nonempty, closed, and disjoint sets, with B as bounded.(d) A and B are nonempty, disjoint, and compact sets.

32. Let∑an be a series of nonnegative terms. Which of the following,

if true, ensures the convergence of∑an?

(a) The partial sum of∑an form a bounded sequence.

(b) (an)→ 0 as n→∞.(c) (an)→ 1 as n→∞.(d) ( an

an+1)→ 1 as n→∞.

33. The series∑

n3

3nxn converges, if

(a) x = 2.(b) x = 3.(c) x = 4.(d) x = 5.

34. Let f : R→ R be defined by

f(x) = limm→∞

limn→∞

| cos(m!πx) |n .

Which of the following is true?(a) limx→a− f(x) exists for some a ∈ R.(b) f is not continuous at any rational number.(c) f is continuous at π.(d) f has jump discontinuities.

35. Pick out a valid statement.(a) There does not exist a sequence in {m

2n: m ∈ Z, n ∈ N}, which converges

to π.

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(b) Every set B ⊆ R contains a countable set A that is dense in B.(c) The sequence {sinn} is divergent.(d) There does not exist a sequence with irrational terms and converging toa rational number.

36. Which of the following is false?(a) If A1 ⊇ A2 ⊇ A3 ⊇ ... is a nested sequence of infinite sets, then ∩∞n=1Anis also infinite.(b) A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C).(c) A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C).(d) A ∩ (B ∩ C) = (A ∩B) ∩ (A ∩ C).

37. The supremum of { nm+n

: m,n ∈ N}(a) is 1.(b) belongs to it.(c) is 1

2.

(d) does not exist.

38. Which of the following series is convergent?(a)

∑1

n2+n.

(b)∑

1n+1

.

(c)∑

1n logn

.

(d)∑

cosn.

39. Let f : R3 → R2 be defined by

f(x, y, z) = (x2 + y2, ex+z).

The derivative matrix at (1, 2, 3) is(a) 2 e4

4 00 e4

(b) 2 e4

4 00 −e4

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(c) [2 4 0e4 0 e4

](d) [

−2 4 0e4 0 −e4

]40. Let (xn), (yn) be sequences in (0,∞). Suppose 0 < y1 < x1. If

xn+1 =xn + yn

2and yn+1 =

√xnyn.

Then,(a) (xn) is decreasing but not bounded below.(b) (yn) is increasing but not bounded above.(c) (xn) and (yn) both converge to the same limit.(d) (xn) and (yn) both converge but to the different limits.

Answer Key

1. a, c2. a, c, d3. a, b, c, d4. a, c, d5. b6. a, c7. a, b, d8. b, c9. a, b, c, d10. b, d11. c12. a, b, d13. a, c, d14. b, d15. a, b, c16. a17. a, b

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18. a, b, c, d19. c, d20. a, c21. a, c, d22. b, c, d23. d24. b, c, d25. a, c, d26. c27. a28. c29. d30. c31. c32. a33. a34. b35. b36. a37. a38. a39. c40. c

Best Wishes by Praveen Chhikara...

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TABLE OF CONTENTS · Real Analysis Test 1 ... Countable sets, Sequences and Series, Point-set topology, Continuity, Uniform Continuity, Differentiability, Uniform Convergence,Riemann

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