US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.1
Continuous Functions on Metric Spaces.
Objective Questions 3.1
. (Revised Syllabus 2018-19)
(1) Let d be the usual distance in R. For any A ⊆ R, dA : R −→ R is defined by dA(x) =
inf{d(x, a) : a ∈ A}. Then
(a) dR, dR\Q are not continuous on R and dQ(x) > 0 ∀x ∈ R \Q.
(b) dQ ≡ 0 and dR\Q ≡ 0 on R and dQ, dR\Q are continuous on R.
(c) dR, dR\Q are continuous on R and dR\Q(x) > 0 ∀x ∈ Q.
(d) None of the above.
(2) Let d denote the usual distance in R and for A ⊆ R, let
χA(x) =
{
1 if x ∈ A
0 if x /∈ A .
Then
(a) χA is continuous on R if and only if A is an open subset of R.
(b) χA is continuous on R if and only if A is a closed subset of R.
(c) χA is continuous on R if and only if A = ∅ or A = R.
(d) None of the above.
(3) Consider the metrics d and d1 on N, where d is the induced distance from R with usual
distance and d1(m,n) =
∣∣∣ 1m − 1n ∣∣∣ for m,n ∈ N. Let i : N −→ N denote the identity map on
N. Then
(a) i : (N, d) −→ (N, d1) is continuous but i : (N, d1) −→ (N, d) is not continuous.
(b) i : (N, d) −→ (N, d1) is not continuous.
(c) i : (N, d1) −→ (N, d) is not continuous.
(d) None of the above.
(4) Let d1 and d2 be equivalent metrics on X and (Y, d) be any metric space. If f : (X, d1) −→
(Y, d) and g : (Y, d) −→ (X, d1) are continuous maps on X and Y respectively, then
(a) f : (X, d2) −→ (Y, d) is continuous, but g : (Y, d) −→ (X, d2) may not be continuous
(b) f : (X, d2) −→ (Y, d) may not be continuous, but g : (Y, d) −→ (X, d2) is continuous
(c) f : (X, d2) −→ (Y, d) and g : (Y, d) −→ (X, d2) are continuous on X and Y respectively.
(d) None of the above.
(5) Let A = {x ∈ R : sinx = 1
2
}, the distance in R being usual. Then
(a) A is an infinite closed set. (b) A is a finite closed set.
(c) A is an open set. (d) None of the above.
(6) Let (X, d) and (Y, d′) be metric spaces and f, g : X −→ Y be continuous maps. If A ⊆ X
such that f(x) = g(x) ∀ x ∈ A, then the statement which is not true is
1
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(a) f(x) = g(x) ∀x ∈ A◦
(b) f(x) = g(x) ∀ x ∈ A
(c) f(x) = g(x) ∀ x ∈ δA where δA is the boundary of A
(d) All the above statements are false
(7) Let (X, d) and (Y, d′) be metric spaces and f, g : X −→ Y be continuous maps. Let
A = {x ∈ X : f(x) = g(x)}. Then
(a) A is a dense subset of X. (b) A is a closed subset of X.
(c) A is an open subset of X. (d) None of the above.
(8) Let d denote the usual distance in R and d1 denote the discrete metric on R. Let i :
(R, d1) −→ (R, d) be the identity map. Then
(a) i(Q) ⊆ i(Q). (b) i−1(Q) ⊆ i−1(Q).
(c) i−1(Q) ⊆ i−1(Q). (d) None of the above.
(9) Let (X, d) and (Y, d′) be metric spaces and f : X −→ Y. Let {An}n∈N be a family of closed
subsets of X. Then the statement which is not true is
(a) If f is continuous on A1 and A2, then f is continuous on A1 ∪ A2.
(b) If f is continuous on each An, then f is continuous on
∞⋃
n=1
An.
(c) If f is continuous on each An, then f is continuous on A =
⋂
n∈N
An, provided A 6= ∅
(d) None of the above.
(10) Let f : R2 −→ R (the distance in R and R2 are Euclidean) be defined by f(x, y) = |x|.
Then
(a) f is not continuous at (x, 0) for each x ∈ Z.
(b) f is not continuous at (0, 0).
(c) f is continuous on R2
(d) None of the above.
(11) f, g : R −→ R are any maps, such that f ◦ g and g ◦ f are continuous (distance being
usual). Then
(a) f : R −→ R and g : R −→ R are continuous
(b) f ◦ g = g ◦ f
(c) At least one of f and g is coninuous.
(d) Neither f nor g may be continuous.
(12) Let (X, d) be a metric space where X is a finite set and (Y, d′) be any metric space. Let
f : X −→ Y. Then the statement which is not true is
(a) f is continuous on X
(b) f(X) is bounded.
(c) If A is open in X, f(A) is open in Y
(d) If B is closed in Y, f−1(B) is closed in X.
2
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(13) Let (X, d) be a compact metric space and f : X −→ (0,∞). (distance usual) be a
continuous function. If inf{f(x) : x ∈ X} = m, then
(a) m may be 0 (b) m = 0 (c) m > 0 (d) m may be negative
(14) Let (X, d) be a finite metric space, |X| > 2. If f : X −→ R (usual distance) is a continuous
function, then
(a) |f(X)| ≥ 2 (b) f(X) = [m,M ] for some m,M ∈ R
(c) f is a constant function. (d) None of the above.
(15) Let (X, d) be a metric space. If f, g ∈ C (X ,R), then
(a) f + g ∈ C (X ,R), but f–g may not be in C (X ,R).
(b) f + g, f − g and 2f ∈ C (X ,R).
(c) f + g, f − g ∈ C (X ,R), but 2f may not be in C (X ,R)
(d) f + g, f − g ∈ C (X ,R), but fg may not be in C (X ,R)
(16) Let X1 = [0, 1];Y1 = [0,∞);X2 = (0, 1) ∪ (2, 3), Y2 = (0, 1);X3 = (0, 1), Y3 = {0, 1}. Then
there exists a continuous onto function from Xi −→ Yi when
(a) i = 1, 2, 3 (b) i = 1, 2 (c) i = 2 (d) i = 3
(17) Consider the map L : C [0 , 1 ] −→ R (usual distance) defined by L(f) =
∫ 1
0
f(t) dt. Then,
(a) L : (C [0 , 1 ], ‖ ‖1 ) −→ R is continuous but L : (C [0 , 1 ], ‖ ‖∞) −→ R is not continuous.
(b) L : (C [0 , 1 ], ‖ ‖∞) −→ R is not continuous.
(c) L : (C [0 , 1 ], ‖ ‖1 ) −→ R and L : (C [0 , 1 ], ‖ ‖∞) −→ R are both not continuous.
(d) None of the above.
(18) Consider the map φ : C [0 , 1 ] −→ R defined by φ(f) = f(0). Then
(a) φ : (C [0 , 1 ], ‖ ‖∞) −→ R is not continuous.
(b) φ : (C [0 , 1 ], ‖ ‖∞) −→ R is continuous.
(c) φ : (C [0 , 1 ], ‖ ‖∞) −→ R and φ : (C [0 , 1 ], ‖ ‖1 ) −→ R are not continuous.
(d) None of the above.
(19) Let (X, d) be a metric space and f ∈ C (X ,R) be a bounded function. Then f
(a) attains both bounds.
(b) attains at least one bound.
(c) may not attain either bound.
(d) None of the above.
(20) Let f : R −→ R be continuous function (distance usual) and A ⊆ R. Consider the
following statements:
(i) If A is closed and bounded, f(A) is closed and bounded.
(ii) If A is closed, f(A) is closed. (iii) If A is bounded, f(A) is bounded.
(a) (i), (ii), (iii) are true statements.
(b) (i) and (iii) are true, (ii) is not true.
(c) Only (i) is true.
(d) (i) and (ii) are true, (iii) is not true.
3
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(21) Let (X, d) be a compact metric space and f : X −→ R is continuous. Let (xn) be a
sequence in X. Which statement is false?
(a) If (xn) is convergent (f(xn)) is convergent.
(b) If (xn) is Cauchy, (f(xn)) is Cauchy.
(c) (f(xn)) has convergent subsequence.
(d) None of (a), (b), (c) are false.
Topology of Metric Spaces and Real Analysis: Practical 3.1
Continuous Functions on Metric Spaces
Descriptive Questions 3.1
(1) Let f, g : R −→ R be continuous function (with respect to usual distance). Let h : R2 −→
R2 be defined by h(x, y) = (f(x), g(y)). Show that h : (R2, d) −→ (R2, d) is Continuous
where d is Euclidean distance.
(2) Let f : R2 −→ R be continuous map. Show that g : R2 −→ R defined by g(x, y) =
f(x+ y, x− y) is continuous.
(3) Show that i : (R, d) −→ (R, d1) where d is usual distance in R and d1 is discrete metric on
R is not continuous where i is the identity map on R.
(4) Let (X, d) be a metric space and let A ⊆ X, If dA : X −→ R is defined by dA(x) = d(x,A).
Show that dA is continuous.
(5) X = M2(R) and ‖A‖ =
( ∑
1≤i,j≤2
a2(ij)
) 1
2
. Show that f : X −→ R (distance usual) defined
by f(A) = detA is continuous. Hence show that
(i) (GL)2(R) is an open subset of X.
ii) (SL)2(R) is a closed subset of X.
(6) Prove or disprove:
a) If (X, d) and (Y, d′) are metric spaces and f : X −→ Y is a continuous bijective map,
then for any open ball B in (X, d), f(B) is an open ball (Y, d′).
b) Let (X, d) and (Y, d′) be metric spaces. If (X, d) is complete and f : X −→ Y is
continuous and onto, then (Y, d′) is complete.
(7) Let (X, d) and (Y, d′) be metric spaces. Prove that f : X −→ Y is continuous on X if and
only if f is continuous on each compact subset of X.
(8) Let A,B be two compact subsets of a metric space (X, d) such that A∩B 6= ∅. Show that
d(A,B) > 0 and ∃ a ∈ A, b ∈ B such that d(A,B) = d(a, b).
(9) Let K ⊆ Rn be such that any continuous function fromK to R be bounded. Show that K
is compact.
4
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(10) Show that S1 = {(x, y) ∈ R2 : x2 + y2 = 1} is a compact subset of R2, distance being
Euclidean.
(11) Let f : X −→ (0,∞) be a continuous function, where (X, d) is a compact metric space.
Show that ∃  > 0 such that f(x) ≥ ,∀x ∈ X.
(12) ψ : (C[0, 1], ‖ ‖∞) −→ R (usual distance) defined by ψ(f) = f(0) is continuous.
(13) L : (C[0, 1], ‖ ‖∞) −→ R (usual distance) defined by L(f) =
∫ 1
0
f(t) dt is continuous.
5
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.2
Uniform Continuity and Fixed Point Theorem
Objective Questions 3.2
. (Revised Syllabus 2018-19)
(1) f : R \ {0} −→ R defined by, f(x) = 1
x
for x 6= 0 is uniformly continuous on
(a) (0, 1) (b) (0,∞) (c) [1,∞) (d) None of these.
(2) f(x) =
1
1 + x2
for x ∈ R is uniformly continuous on
(a) [0, 1] but not on [0,∞)
(b) [1,∞) but not on [0,∞)
(c) R
(d) None of these.
(3) Let A ⊆ R. If f, g : A −→ R are uniformly continuous on A, then
(a) f + g is uniformly continuous on A but f · g may not be uniformly continuous on A.
(b) f + g and f · g are uniformly continuous on A.
(c) Neither f + g nor f · g may be uniformly continuous on A.
(d) None of the above.
(4) Consider the following functions (distance in R is usual):
(i) f : [0, 2pi] −→ R, f(x) = x sinx
(ii) f : (0, 1) −→ R, f(x) = 1
x
(iii) f : [0, 1]× [0, 1] −→ R, f(x, y) = x+ y ( distance in R2 Euclidean)
(a) (i), (ii), (iii) are uniformly continuous.
(b) (i) and (iii) are uniformly continuous, (ii) is not.
(c) Only (i) is uniformly continuous.
(d) Only (iii) is uniformly continuous.
(5) Suppose A and B are closed subsets of R and f : A ∪ B −→ R is uniformly continuous on
A as well as B. Then,
(a) f is uniformly continuous on A ∪B.
(b) f is uniformly continuous on A ∪B if A ∩B = ∅.
(c) f may not be uniformly continuous on A ∪B.
(d) None of the above.
(6) f : [0,∞) −→ R defined by f(x) = √x is
(a) continuous on [0,∞) but not uniformly continuous on [0,∞).
(b) uniformly continuous on [0, 1] but not on [0,∞).
(c) uniformly continuous on [0,∞).
(d) None of the above.
6
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(7) If f, g : R −→ R are uniformly continuous on R, then
(a) The product f · g uniformly continuous on R.
(b) The composites f ◦ g and g ◦ f uniformly continuous on R.
(c) f 2 and g2 are uniformly continuous on R.
(d) None of the above.
(8) Let (X, d) be a metric space and A be a non-empty subset of X. Then dA : X −→ R
defined by dA(x) = d(x,A) = inf{d(x, a) : a ∈ A} is
(a) continuous on A but not on X.
(b) uniformly continuous on X.
(c) not uniformly continuous on X.
(d) None of these.
(9) Let (X, d) and (Y, d′) be a metric spaces and f : X −→ Y . Suppose (xn) is a Cauchy
sequence in X, then {f(xn)} is a Cauchy sequence in Y if
(a) f is continuous on X.
(b) f is uniformly continuous on X.
(c) X and Y are complete.
(d) None of these.
(10) Let A ⊆ R, A is bounded but not closed. Then
(a) Any continuous function from A to R is bounded.
(b) Any continuous function from A to R is uniformly continuous.
(c) Any continuous, bounded function from A to R attains bounds.
(d) None of the above.
(11) Let (X, d) and (Y, d′) be metric spaces and f : X −→ Y a uniformly continuous function.
Then the statement which is not true is
(a) Given a bounded subset A of X, f(A) need not be a bounded subset of Y .
(b) If {xn} is a Cauchy sequence in X, then {f(xn)} is a Cauchy sequence in Y .
(c) If {f(xn)} is a Cauchy sequence in Y , {xn} is a Cauchy sequence in X.
(d) If {xn} is convergent, then {f(xn)} is convergent.
(12) Consider the following maps:
(i) f : R −→ R such that f is differentiable and |f ′(x)| ≤M ∀ x ∈ R.
(ii) A linear transformation T : Rn −→ Rm.
(iii) A map f : R −→ R satisfying Lipchitz condition namely ∃ M ≥ 0 such that |f(x) −
f(y)| ≤M |x− y| ∀ x, y ∈ R.
Then
(a) (i) and (iii) are uniformly continuous.
(b) (i), (ii) and (iii) are uniformly continuous.
(c) only (iii) is uniformly continuous.
(d) None of above.
(13) Which of the following real valued functions are uniformly continuous on the give sets.
7
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(i) f(x) =
1
x
on (0, 1). (ii) f(x) = x
1
3 on [0, 1].
(a) Only (i) (b) only (ii) (c) both (i) and (ii) (d) Neither (i) nor (ii).
Topology of Metric Spaces and Real Analysis: Practical 3.2
Uniform Continuity and Fixed Point Theorem
Descriptive Questions 3.2
(1) Show that the function f(x) =
1
1 + x2
for x ∈ R is uniformly continuous on R.
(2) Prove or disprove:
If f, g : R −→ R are uniformly continuous on a nonempty set A ⊆ R then the product
function f · g is uniformly continuous on A.
(3) If f : R −→ R is such that f ′(x) exists ∀ x ∈ R and ∃ a constant M such that |f ′(x)| ≤
M ∀ x ∈ R, then show that f is uniformly continuous on R.
(4) If (X, d), (Y, d′) are metric spaces, then prove that any Lipschitz function f : (X, d) −→
(Y, d) is uniformly continuous. Hence, deduce that sin x, cosx are uniformly continuous on
R.
(5) Let A = (0, 1] ⊂ R. Define dA : A˚ → R as dA(x) = d(x,A). Draw graph of dA. Further,
prove that if (X, d) is a metric space and A ⊆ X then dA : X −→ R defined by dA(x) =
d(x,A) for x ∈ X is uniformly continuous on X.
(6) Let f : [a, b] −→ [a, b] be differentiable and |f ′(x)| ≤ c with 0 < c < 1. Then show that f
is a contraction of [a, b].
(7) Let X and Y be metric spaces. Assume that Y is a discrete metric space and that f :
X −→ Y is a contraction. What can you conclude about f?
(8) Define a sequence of positive real numbers by letting x0 to be any positive real number and
xn+1 = (1 + xn)
−1. Show that this sequence converges and find its limit. (Hint: Prove that
f is a contraction mapping where f : [x0,∞)→ R defined as f(x) = 1
1 + x
).
8
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.3
Connected Sets , Connected Metric Spaces
Objective Questions 3.3
. (Revised Syllabus 2018-19)
(1) Let (X, d) be a discrete metric space
(a) X is connected.
(b) X is connected only if X is infinite.
(c) X is connected if and only if X is a singleton set.
(d) None of above.
(2) Let d be usual distance in R and d1 be the discrete metric in R. Then
(a) [0, 1] is a connected subset (R, d) as well as (R, d1).
(b) [0, 1] is connected subset of (R, d) but not connected subset of (R, d1).
(c) [0, 1] is not a connected subset of (R, d) but a connected subset of (R, d1)
(d) [0, 1] is not a connected subset of (R, d) as well as (R, d1)
(3) If A is a connected subset of (R, d) ( d being usual distance) then
(a) A◦ and A are connected.
(b) A◦ may not be connected but A is connected.
(c) Both A◦ and A may not be connected.
(d) A◦ is connected, but A may not be connected.
(4) Let A,B be connected subsets of (R, d) where d is the usual distance in R. If A ∩ B 6= ∅,
then the following set may not be connected.
(a) A ∪B (b) A ∩B (c) A \B (d) A×B in R2 ( Euclidean distance).
(5) Let A ⊆ Q. If A is a connected subset of (R, d) where d is usual distance then
(a) A = Q (b) A is an infinite bounded set.
(c) A is a singleton set. (d) None of the above.
(6) Consider the following subsets of (R2, d) where d Euclidean .
(i){(x, y) ∈ R2 : xy = 1} (ii){(x, y) ∈ R2 : x = 0}
(iii){(x, y) ∈ R2 : xy = 0} Then,
(a) (i), (ii)(iii) are all connected. (b) (ii), (iii) are connected.
(c) Only (iii) is connected. (d) Only (i) is connected.
(7) Let A,B be non-empty closed subsets of a metric space (X, d). If A ∪ B and A ∩ B are
connected subsets of X, Then6 (a) A and B are both connected.
(b) A and B are both not connected.
(c) A and B are connected if and only if A = B
(d) None of these.
(8) Let (X, d) be a finite metric space. If A ⊆ X is connected then
(a) A = X (b) A 6= X (c) A is a singleton set. (d) A has more than one element.
9
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(9) If A,B are connected subsets of (R2, d) where d is usual distance and A ∩B 6= ∅, then
(a) A ∪B is connected but A ∩B may not be connected.
(b) A ∪B may not be connected but A ∩B is connected.
(c) A ∪B and A ∩B are connected.
(d) None of the above.
(10) Consider (R2, d) where d is Euclidean metric and A be an open ball in R2 and L be a line
in R2. Then
(a) A ∪ L is connected if L does not intersect A.
(b) A ∪ L is connected if L intersects A.
(c) A ∪ L is disconnected if L intersects A but is not a tangent to A.
(d) Cannot say.
(11) In (R2, d) where d is Euclidean distance, the following set is not connected.
(a) R2 \Q×Q.
(b) R2 \ {(0, 0)}
(c) R2 \ {(x, y) : y = 0}
(d) None of the above.
(12) If A,B are connected subsets of (R, d) where d is usual and A ∩B 6= ∅, then
(a) A ∪B is connected but A ∩B may not be connected.
(b) A ∪B may not be connected but A ∩B is connected.
(c) A ∪B and A ∩B are connected.
(d) None of the above.
(13) Let A and B be connected subsets in a metric space (X, d) and A ⊆ C ⊆ B Then,
(a) C is connected .
(b) C◦ is connected.
(c) C is connected.
(d) C ∩ A is connected.
Topology of Metric Spaces and Real Analysis: Practical 3.3
Connected Sets , Connected Metric Spaces
Descriptive Questions 3.3
(1) Let (X, d) be a metric space and A,B ⊆ X be closed. Prove that A ∩ Bc and B ∩ Ac
separated.
(2) Let (X, d) be a metric space and A,B,C ⊆ X. If A and B are separated, B and C are
separated, then prove that A ∪ C and B are separated.
(3) Find the components of the followings:
(i) [0, 1] ∪ [2, 3] with usual distance.
(ii) (0, 1) ∪ {2, 3} with usual distance .
(iii) Q
10
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(iv) R \Q
(v) [0, 1] with distance metric.
(vi) {1, 2, 3} with any metric.
(vii)N with usual distance .
(viii) {(x, y) ∈ R2 : x ∈ Q or y ∈ Q} with Euclidean distance in R2.
(4) Find the connected subsets of the following metric spaces:
(i) (X, d) where d is discrete metric.
(ii) (X, d) where X is a finite set.
(iii)(N, d) where d is usual distance in R.
(iv) (Q, d) where d is usual distance in R
(5) Show that the following subsets of (R2, d) (d being Euclidean distance) are not connected.
(i) {(x, y) ∈ R2 : x2 − y2 = 1}
(ii) {(x, y) ∈ R2 : y 6= 0}
(iii) R2 \ {(x, y) ∈ R2 : y = 6}
(6) Prove or disprove:
(i) If A,C are connected subsets of a metric space and A ⊆ B ⊆ C, then B is connected.
(ii) If A◦ and ∂A are connected then A is connected.
(iii) If A,B are connected then A ∪B,A ∩B are connected.
(iii) An open ball in a metric space is connected.
(iv) If A is a connected subset of a metric space (X, d), then A◦ and ∂A ( boundary of A)
are connected.
11
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.4
Path Connectedness, Convex sets, Continuity and Connectedness
Objective Questions 3.4
. (Revised Syllabus 2018-19)
(1) Let (X, d) be a connected metric space. If f : X −→ R (d usual) is a non-constant
continuous function .Then, f(X) is
(a) finite set (b) countable set. (c) singleton set (d) uncountable set.
(2) The unit circle S1 = {x ∈ R2 : ‖x‖ = 1} is (distance Euclidean)
(a) Compact and Connected
(b) Compact but not Connected
(c) Connected but not Compact
(d) neither Compact nor Connected
(3) Let (X, d) be a finite metric space ,|X| ≥ 2. If f : X −→ R (usual distance) is a continuous
function, then
(a) |f(X)| ≥ 2
(b) f(X) is connected.
(c) If f(X) is connected then f is a constant function.
(d) None of these.
(4) Let A be a non-empty connected subset of R2 (distance Euclidean). Let S = {‖a‖ : a ∈ A}.
If every element in S is a rational number then
(a) A is a singleton set.
(b) Each point in A lies on a circle Cr where Cr = {(x, y) ∈ R2 : x2 + y2 = r2} for some
r ∈ Q
(c) Each point in A lies on a parabola x2 = ry for some r > 0.
(d) None of the above.
(5) Let (X, d) be a connected metric space and A ⊆ X consider χA : X −→ R defined by
χA(x) =
{
1 if x ∈ A
0 if x /∈ A
(a) If χA is continuous on X, then A is a finite set.
(b) If χA is continuous on X, then A = ∅ or A = X.
(c) If χA is continuous on X then A is a non-empty proper subset of X.
(d) None of the above.
(6) If f : [a, b] −→ R is a continuous function, then f([a, b]) is
(a) (0,M ] for some M > 0 (b) (m,M) for some m,M ∈ R
(c) [m,M ] for m,M ∈ R (d) None of these.
12
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(7) Which of the following statements is false in Rn ?
(a) Continuous image of a compact set is compact.
(b) Continuous image of a connected set is connected.
(c) Continuous image of a path connected set is path connected.
(d) None of the above.
(8) Let (X, d) be a connected metric space which is not bounded. Let x0 ∈ X and
Ar = {x ∈ X : d(x, x0) = r}(r > 0). Then
(a) Ar = ∅ except for finitely many positive real number r.
(b) Ar 6= ∅ ∀ r > 0
(c) Ar = ∅ ∀ r > 0
(d) None of these.
(9) Let (X, d) be a connected metric space and f : X −→ Z be a continuous map. Then
(a) f is onto . (b) f is one-one.
(c) f is bijective. (d) f is constant
(10) Rn \ {0Rn} is not path connected if
(a) n = 3, (b) n = 4 (c) n = 1 (d) None of these.
(11) In (R2, d) (d Euclidean distance), the following set is not path connected.
(a) R2 \Q×Q
(b) R2 \ {(0, 0)}
(c) R2 − {(x, y) : y = 0}
(d) B((0, 0), r) \ {(0, 0)}
(12) In (R2, d) (d Euclidean distance), the following set is path connected.
(a) B((0, 0), 1) ∪ {(x, y) ∈ R2 : y = 1}
(b) B((0, 0), 1) ∪ {(x, y) ∈ R2 : y = 2}
(c) B((0, 0), 1) ∪ {(x, y) ∈ R2 : x = 2}
(d) None of the above.
(13) Which of the following statements is false:
(a) A path connected subset of Rn (distance being Euclidean) is connected.
(b) A connected subset of Rn (distance being Euclidean) is path connected.
(c) Union of two path connected subsets A,B in Rn distance being Euclidean such that
A ∩B 6= ∅ is again path connected.
(d) If A,B are two path connected subsets of Rn (distance being Euclidean) such that
A ∩B 6= ∅ then A ∩B is path connected.
(14) Let (X, d) and (Y, d′) be metric spaces. If f : (X, d) −→ (Y, d′) is a continuous function,
then
(a) Number of components of (X, d) ≤ Number of components of (Y, d′).
(b) Number of components of (X, d) ≥ Number of components of (Y, d′).
(c) Number of components of (X, d) = Number of components of (Y, d′).
(d) Cannot say.
13
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(15) Let (X, d) and (Y, d′) be metric spaces and f : (X, d) −→ (Y, d′) be a bijective continuous
function, then
(a) Number of components of (X, d) ≤ Number of components of (Y, d′).
(b) Number of components of (X, d) ≥ Number of components of (Y, d′).
(c) Number of components of (X, d) = Number of components of (Y, d′).
(d) Cannot say.
Topology of Metric Spaces and real Analysis: Practical 3.4
Path Connectedness, Convex sets, Continuity and Connectedness
Descriptive Questions 3.4
(1) Prove that the following subsets of Rn (distance being Euclidean) are convex and hence
path connected. (i) an open ball (ii) a closed ball (iii) a line
(2) Let (X, d) be a metric space and A be a proper non-empty subset of X. If the characteristic
function χA is continuous on X, show that X is not connected.
(3) Show that Br((0, 0)) \ {(0, 0)} is path connected in R2 with Euclidean distance.
(4) Show that R2 \ S × S where S is any countable subset of R is path connected. (Hint: For
any x, y ∈ R2 \ S × S there are uncountable lines passing through x and y).
(5) Prove or disprove :
(a) If A is a path connected subset of Rn (distance being Euclidean) thenA◦ is path
connected.
(b) If {An}n∈N is a sequence of path connected subsets of R2 (distance being Euclidean)
such that An+1 ⊆ An ∀n ∈ N and ∩n∈NAn 6= ∅ then ∩n∈NAn is connected.
(6) (a) If (X, d) is a connected metric space and f : X −→ Z a continuous function, prove
that f is constant.
(b) If (X, d) is a connected metric space and (Y, d′) is any metric space, Y being a finite
set, then show that any continuous function f : X −→ Y is constant.
(c) Let (X, d) be a connected metric space and (Y, d1) be a discrete metric space. Show
that any continuous function f : X −→ Y is constant.
(7) Let (X, d) be a connected metric space which is not bounded. Prove that for each x0 ∈ X
and each r > 0, the set {x ∈ X : d(x, x0) = r} is non-empty.
(8) Give an example of a subset of Rn (distance being Euclidean) which is connected but not
path connected.
(9) Show that if (X, d) is a connected metric space then either X is countable or X is singleton.
(10) Show that the following sets are path connected subsets of R2.
14
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(i) E = {(x, y) ∈ R2, x > 0, x2 − y2 = 1}
(ii) Er = {(x, y) ∈ R2 : x2 + y2 = r2}
(iii) E = {(x, y) ∈ R2 : xy = 0}
(iv) E = {(x, y) ∈ R2 : y2 = x} ∪ {(x, y) ∈ R2 : y2 = −x}
(v) E = {(x, y) ∈ R2 : y = 0}
(vi) E = {(x, y) ∈ R2 : 1 < 2x+ y < 3}
(vii) S1 = {(x, y) ∈ R2 : x2 + y2 = 1}
(viii) E = {(x, y) ∈ R2 : 0 < x < 2, 1 < y < 5}
15
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.5
Pointwise and Uniform Convergence of Sequences of Functions and Properties
Objective questions 3.5
. (Revised Syllabus 2018-19)
(1) χn : R −→ R χn(x) =
{
1 if x ∈ [−n, n]
0 if x /∈ [−n, n]
(a) {χn} converges pointwise to 0 on R.
(b) {χn} does not converge uniformly on R.
(c) {χn} converges uniformly to 1 on R.
(d) None of the above.
(2) Let fn(x) = sinnx for x ∈ R. and gn(x) = fn(x)
n
∀x ∈ R. Then
(a) {fn} and {n} are uniformly convergent on R.
(b) {fn} and {n} are not pointwise convergent on R.
(c) {gn} is uniformly convergent on R but {fn} is not.
(d) {fn} is uniformly convergent on R but {gn} is not.
(3) Let fn : [0, 1] −→ [0, 1] be defined by fn(x) = x ∗ χn(x) where χn(x) =

0 if x /∈
[
0,
1
n
]
1 if x ∈
[
0,
1
n
]
(a) {fn} converges uniformly to 0 on [0, 1].
(b) {fn} converges pointwise to 1 on [0, 1] but does not converge uniformly.
(c) {fn} converges uniformly to 1 on [0, 1].
(d) None of the above.
(4) The least integer value of k for which
{
e−nx
nk
}
is uniformly convergent on [0,∞) is
(a) 0 (b) 1 (c) −1 (d) 2
(5) If {fn} and {gn} are sequences of functions on S, S ⊆ R converging uniformly to f and g
respectively on S then the following sequence of functions may not converge uniformly of
S to the given function.
(a) {fn + gn} to f + g. (b) {fn − gn} to f − g. (c) {λfn} to λf. (d) {fn ∗ gn} to
f ∗ g.
(6) Let fn(x) =
xn
1 + xn
, 0 ≤ x ≤ 1.
(a) {fn} converges uniformly on [0, 1]
(b) {fn} converges uniformly on
[
1
2
, 1
]
16
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(c) {fn} converges uniformly on
[
0,
1
2
]
(d) {fn} converges uniformly on (0, 1]
(7) Let fn(x) =
xn
n
∀ x ∈ [0, 1]. Then
(a) {fn} converges uniformly to 0 but f ′n does not converge uniformly on [0, 1].
(b) {fn} converges uniformly to 0 and f ′n converges uniformly to 1 on [0, 1].
(c) {fn} does not converges uniformly on [0, 1] but f ′n converges uniformly on [0, 1].
(d) None of the above.
(8) Let fn(x) =
x
x+ n
for x ∈ [0,∞). Show that {fn} does not converge uniformly on [0,∞)
but converges uniformly on [0, a] where a > 0. Also show that {fn} does not converge
uniformly on [a,∞], a > 0
(a) {fn} converges uniformly on [0,∞)
(b) {fn} converges uniformly on [a,∞), a > 0
(c) {fn} converges uniformly on [0, a], a > 0
(d) None of the above.
(9) gn(x) = x
n−1(1− x), 0 ≤ x ≤ 1.
(a) {gn} is uniformly convergent on [0, 1].
(b) {gn} is not uniformly convergent on [0, 1].
(c) {gn} is not pointwise convergent on [0, 1].
(d) None of the above.
(10) {fn} and {gn}, gn 6= 0 are real valued functions on a non-empty subset S, S ⊆ R which
are uniformly convergent to the functions f and g respectively on S.
(a) {fn ∗ gn} need not be uniformly convergent on S.
(b) {fn/gn} is uniformly convergent on S.
(c) {fn ∗ gn} is uniformly convergent to f ∗ g on S if each fn is bounded on S.
(d) {fn ∗ gn} converges uniformly to f ∗ g on S if and only if either f ≡ 0 or g ≡ 0 on S.
(11) Let {fn} be a sequence of real valued functions on a set S converging uniformly to a
function f . Then the following statement is not true.
(a) Each fn is bounded on S =⇒ f is bounded on S.
(b) Each fn is differential on S =⇒ f is differentiable on S
(c) Each fn is continuous on S =⇒ f is continuous on S.
(d) Each fn is integrable on S =⇒ f is integrable on S.
(12) Let {fn} be a sequence of real valued R−integrable functions on [a, b] and f be the
pointwise limit of {fn}
(a) If lim
n−→∞
∫ b
a
fn 6=
∫ b
a
f then {fn} doesn’t converge uniformly to f .
(b) If {fn} doesn’t converge uniformly to f , then lim
n−→∞
∫ b
a
fn 6=
∫ b
a
f.
17
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(c) If lim
n−→∞
∫ b
a
fn 6=
∫ b
a
f then the convergence is uniform.
(d) None of the above.
(13) Let {fn} be a sequence of differentiable functions on (a, b). Let lim
n−→∞
fn(x) = f(x), lim
n−→∞
f ′n(x) =
g(x) (pointwise limits)
(a) If f is differntable on (a, b), then f ′ = g on (a, b)
(b) If {f ′n} converges uniformly to g, then f is differentable on (a, b) and f ′ = g.
(c) If f ′ = g on (a, b) then {fn} converges uniformly to f on (a, b).
(d) If {fn} converges uniformly to f, then f is differentiable and f ′ = g on (a, b)
(14) Let fn(x) =
{
x if x ≤ n
n if x ≥ n
(a) {fn} converges uniformly on R to a bounded function.
(b) {fn} converges uniformly on R to an unbounded functions.
(c) {fn} is not pointwise convergent on R.
(d) {fn} converges pointwise on R.
(15) Let fn(x) =
x
1 + nx2
(a) {fn} converges uniformly on R but {f ′n} does not converge uniformly on R.
(b) {fn} converges uniformly on R and {f ′n} also converges uniformly on R.
(c) {fn} does not converge uniformly on R but {f ′n} converges uniformly on R.
(d) Neither {fn} nor {f ′n} converge uniformly on R.
(16) Let fn(x) =
xn
1 + xn
on [0, 2] and f(x) = lim
n−→∞
fn(x)
(a) {fn} converges uniformly to f on [0, 2] and f is continuous at x = 1.
(b) {fn} does not converge uniformly to f on [0, 2] and f is not continuous on [0, 1].
(c) {fn} does not converge uniformly to f on [0, 2] but f is continuous on [0, 2].
(d) None of the above.
(17) fn(x) = x
n for x ∈ [0, 1]
(a) The pointwise limit of {fn} is not continuous on [0, 1]
(b) {fn} converges pointwise on [0, 1] to a continuous function.
(c) {fn} converges uniformly on [0, 1] to a continuous function.
(d) None of the above.
(18) fn(x) =
xn
n
for x ∈ [0, 1]. Let lim
n−→∞
fn(x) = f(x), lim
n−→∞
f ′n(x) = g(x)
(a) {fn} and {f ′n} converge uniformly on [0, 1].
(b) {f ′n} converges uniformly to g on [0, 1]
(c) {f ′n} does not converge uniformly to g on [0, 1].
(d) None of the above
18
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.5
Pointwise and Uniform Convergence of Sequences of Functions and Properties
DESCRIPTIVE QUESTIONS 3.5
(1) Show that each of the following sequences of functions converges pointwise on (0, 1). Identify
the subintervals on which the convergence is uniform.
(i)
n
nx+ 1
(ii)
x
nx+ 1 (iii)
1
nx+ 1
(2) Examine the following sequences of functions for pointwise and uniform convergence on
[0, 1]
(i) nxe−nx
2
(ii) n
1
2x(1− x2)n (iii) nx(1− x2)n2
(3) Examine the following sequences of functions for pointwise and the uniform convergence
on [0,∞). In case of the convergence not being uniform, examine whether the convergence
is uniform on [0, a] or [a,∞) where a > 0.
(i) e−nx (ii)
sinnx
1 + nx
(iii) x2e−nx
(iv)
xe
−x
n
n
(v) n2x2e−nx
(4) fn : (0,∞) −→ R, fn(x) = n
1 + nx
. Then
(i) Show that {fn} is bounded on (0,∞) for each n ∈ N.
(ii) Find the pointwise limit f of {fn} and show that f is not bounded on (0,∞).
(iii) Is {fn} uniformly convergent on (0,∞)? State clearly the theorem you used.
(iv) Show that there does not exist α ∈ R+ such that |fn(x)| ≤ α for all n ∈ N and for all
x ∈ (0,∞).
(5) Show that the following sequences of functions do not converge uniformly on the given
domain.
(i) fn : [0,∞) −→ R, fn(x) =
{
x if x ≤ n
n if x > n
(ii) fn : [0,∞), fn(x) = nx
1 + nx2
.
(iii) fn : (0, 1] −→ R, fn(x) =
0 if 0 < x ≤
1
n
1
x
if 1
n
< x ≤ 1
19
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(6) fn : [0, 1] −→ R, fn(x) = nxe−nx. Show that each fn continuous on [0, 1], the pointwise
limit of {fn} continuous on [0, 1] but {fn} does not converge uniformly convergent on [0, 1].
(7) Show that the following sequence of functions do not converge uniformly on the given
domain.
fn : [0, 1] −→ R, fn(x) =
{
nx if 0 ≤ x ≤ 1
n
1 otherwise
.
(8) Let fn : [0, 1] −→ R, fn(x) = 1
nx+ 1
.
Show that {fn} converges pointwise to f on [a, b] and each fn and f are R−integrable on
[0, 1] with lim
n−→∞
∫ b
a
fn(x) dx =
∫ b
a
f(x)dx but {fn} does converges uniformly on [0, 1].
(9) Let fn : [0, 1] −→ R, fn(x) =
{
n2 if 0 < x < 1
n
0 otherwise
. Show that {fn} does not converge
uniformly on [0, 1]. (Hint: show that if each fn is R-integrable on [0, 1] and fn −→ f
pointwise on [0, 1] but lim
n−→∞
∫ b
a
fn(x) is not convergent.)
(10) Let fn : [0, 1] −→ R is defined for n ≥ 2, fn(x) =

n2x if 0 ≤ x ≤ 1
n
−n2
(
x− 2
n
)
if 1
n
≤ x ≤ 2
n
0 if 2
n
≤ x ≤ 1
. Show
that {fn} does not converge uniformly on [0, 1].
(Hint: Show that each fn is R-integrable on [0, 1] and fn −→ f pointwise on [0, 1] , but
lim
n−→∞
∫ 1
0
fn(x) 6=
∫ 1
0
f(x)dx.)
(11) Let fn : [−1, 1] −→ R, fn(x) =
√
x2 + 1
n2
. Given that fn −→ f uniformly on [−1, 1] where
f(x) = |x| for x ∈ [−1, 1]. Find lim
n−→∞
∫ 1
−1
fn(x) dx.
(12) Let fn : [0, 1] −→ R, fn(x) = x + n. Does {fn} converge pointwise at any x ∈ [−1.1].
Does sequence {fn} converge uniformly on [−1, 1]? Show that {f ′n} converges uniformly on
[−1, 1].
(13) fn : R −→ R, fn(x) = e
−n2x2
n
. Find the pointwise limit function f of {fn} and g of {f ′n}.
Does f ′n −→ g uniformly on R?. Is f ′(0) = g(0)?
20
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.6
Objective Questions 3.6
. (Revised Syllabus 2018-19)
(1) The series
∞∑
n=1
nx2
n3 + x3
is
(a) uniformly convergent on [0, A] where A > 0 but not on [0,∞).
(b) not uniformly convergent on [0, A] where A > 0.
(c) uniformly convergent on [0,∞).
(d) none of the above.
(2) The series
∞∑
n=1
xn
n+ 1
is
(a) uniformly convergent on R.
(b) not uniformly convergent on [−a, a] where 0 < a < 1
(c) uniformly convergent on [−a, a] where 0 < a < 1.
(d) none of the above.
(3) The series
∞∑
n=1
xn
xn + 1
is
(a) pointwise convergent on [1,∞).
(b) uniformly convergent on [0, a], a < 1.
(c) uniformly convergent on [0,∞).
(d) none of the above.
(4) The series
∞∑
n=1
x
[(n− 1)x+ 1)][nx+ 1] is
(a) uniformly convergent on [0,∞).
(b) uniformly convergent on [0, 1].
(c) uniformly convergent on [a, b], a > 0.
(d) none of the above.
(5) The series
∞∑
n=1
(−x)n(1− x) is
(a) uniformly convergent on R.
(b) uniformly on [0, 1].
(c) uniformly convergent on [0, a] where 0 ≤ a < 1 but not on [0, 1].
(d) none of the above.
(6) The least value of integer k for which
∞∑
n=1
sinnx
nk
converges uniformly on R is
(a) 1. (b) 2. (c) −1. (d) none of the above.
21
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(7)
∞∑
n=1
|an| is convergent then
∞∑
n=1
anx
n is
(a) uniformly convergent on R.
(b) uniformly convergent on any bounded interval.
(c) uniformly convergent on [−a, a], where 0 ≤ a < 1.
(d) none of the above.
(8) The series
∞∑
n=1
xn(1− x)
(a) converges uniformly to x on [0, a], where 0 ≤ a < 1.
(b) converges uniformly on [0, 1).
(c) is not pointwise convergent at x = 1.
(d) none of the above.
(9) The series
∞∑
n=1
x2
(1 + x2)n
(a) converges uniformly on (0,∞).
(b) converges uniformly on [a,∞), a > 0.
(c) does not converge uniformly on [a,∞), a > 0.
(d) none of the above.
(10) The series
∞∑
n=1
1
(nx)2
(a) converges uniformly on R \ {0}.
(b) does not converge uniformly on [a,∞), a > 0.
(c) converges uniformly on [a,∞), a > 0.
(d) none of the above.
(11) Consider the series
∞∑
n=1
xn(1− 2xn). Then
(a)
∫ 1
0
∞∑
n=1
xn(1− 2xn)dx 6=
∞∑
n=1
∫ 1
0
xn(1− 2xn)dx.
(b)
∫ 1
0
∞∑
n=1
xn(1− 2xn)dx =
∞∑
n=1
∫ 1
0
xn(1− 2xn)dx.
(c) it converges uniformly on [0, 1] and can be integrated term by term.
(d) none of the above.
(12) Let f(x) =
∞∑
n=1
cosnx
n2
. Then
(a) None of the below statements are true.
(b)
∞∑
n=1
cosnx
n2
is not uniformly convergent on [0, 1] and cannot be integrated term by term.
22
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(c)
∞∑
n=1
cosnx
n2
is uniformly convergent on [0, 1] and can be integrated term by term.
(d)
∞∑
n=1
cosnx
n2
is not uniformly convergent on [0, δ], where 0 ≤ δ < 1 and
lim
δ−→1
lim
n−→∞
∫ δ
0
n∑
k=1
cos kx
k2
dx 6= lim
δ−→1
∫ δ
0
∞∑
n=1
cosnx
n2
dx
(13) The power series expansion for
∫ x
0
e−t
2
dt is
(a)
∞∑
n=0
x2n+1
(2n+ 1)!
(b)
∞∑
n=0
(−1)nx2n+1
n!(n+ 1)
(c)
∞∑
n=0
(−1)nx2n+1
(n+ 1)!
(d)
∞∑
n=0
(−1)nx2n+1
(2n)!(n+ 1)
(14) If R is the radius of convergence of power series
∞∑
n=0
cnx
n, then radius of convergence of
the power series
∞∑
n=0
cknx
nk is (a) Rk (b) R (c) R
1
k (d)
1
Rk
(15) If R is the radius of convergence of power series
∞∑
n=0
cnx
n, then radius of convergence of
the power series
∞∑
n=0
cnx
nk is (a) Rk (b) R (c) R
1
k (d)
1
Rk
(16) If R is the radius of convergence of power series
∞∑
n=0
cnx
n then the radius of convergence
of
∞∑
n=0
(−1)n
n2
cnx
n is (a) R2 (b) R (c) 0 (d) ∞
(17)
∞∑
n=0
anx
n has radius of convergence R1 and
∞∑
n=0
bnx
n, has radius of convergence R2.
Let Cn =
{
an if n is even
bn if n is odd
. Then the radius of convergence of the power series
∞∑
n=0
cnx
n
is
(a) R1 +R2 (b) min{R1, R2} (c) max{R1, R2} (d) None of the above.
(18) Let R be the radius of convergence of power series
∞∑
n=0
cnx
n, then the following powe series
does not have radius of convergence R.
(a)
∞∑
n=0
(−1)ncnxn (b)
∞∑
n=0
cn
n
xn (c)
∞∑
n=0
(−1)nc2nxn (d)
∞∑
n=0
(−1)nncnxn
23
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(19) Let
∞∑
n=0
cnx
n be a power series with integer coefficients such that cn 6= 0 for infinitely many
n. If R is the radius of convergence of
∞∑
n=0
xn, then
(a) R = 0 (b) R =∞ (c) R ≤ 1 (d) R ≥ 1
(20) Let f(x) =
∞∑
n=0
cnx
n for |x| < R. If f(x) is an even function for |x| < R, then
(a) cn = 0 ∀ n ∈ N.
(b) cn = 0 when n is even.
(c) cn = 0 when n is odd.
(d) None of the above.
(21) If
∞∑
n=0
cnx
n has radius of convergence 1, then
(a) the power series converges at x = 1 and x = −1.
(b) the power series diverges at x = 1 and x = −1.
(c) the power series converges at x = 1 and diverges at x = −1.
(d) none of the above.
(22) Consider the power series
∞∑
n=0
cnx
n, for which cn =

1
2k
if n = 2k
3k+1 if n = 2k + 1
.
Then the radius of convergence of
∞∑
n=0
cnx
n is
(a) 2 (b)
√
2 (c)
1√
3
(d)
√
3
(23) If α is a non-zero real number then the radius of convergence of αnxn is
(a) |α| (b) 1|α| (c) 0 (d) ∞
(24) If α and β are real numbers such that 0 < |β| < |α| then radius of convergence of
∞∑
n=0
(αn + βn)xn is (a) |α| (b) 1|α| (c) |β| (d)
1
|β|
(25) The series expansion log(1 + x) = x− x
2
2
+
x3
3
− x
4
4
+ · · · is valid if
(a) |x| ≤ 1 (b) |x| ≤ A for A > 0 (c) |x| < 1 (d) x > 0
(26) The series expansion 1 + 2x+ 3x2 + · · ·+ nxn−1 + · · · = 1
(1− x)2 is valid in
(a) R (b) (−1, 1) (c) [−1, 1) (d) [a, b] for any a, b ∈ R, a < b
24
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(27) Let E(x) = 1 + x+
x2
2!
+
x3
3!
+ · · ·+ x
n
n!
+ · · · for x ∈ R. Then lim
x−→∞
xnE(−x) =
(a) 1 (b) 0 (c) ∞ (d) −1
(28) Let E(x) =
∞∑
n=0
xn
n!
, C(x) =
∞∑
n=0
(−1)n x
2n
(2n)!
, S(x) =
∞∑
n=0
(−1)nx2n+1
(2n+ 1)!
, x ∈ R. Then
(a) E(x), C(x), S(x) are one-one
(b) Only E(x) is one-one
(c) C(x), S(x) are one-one
(d) None of the above.
(29) Let L : (0,∞) −→ R be defined as L(E(x)) = x and E(L(y)) = y, x ∈ R.
(a) L(1− y) = −
∞∑
n=1
yn
n
(b) L(y) =
∞∑
n=1
∞∑
n=0
nyn
(c) L(y) =
∞∑
n=1
yn
n+ 1
(d) None of the above.
(30) Let L : (0,∞) −→ R be defined as L(E(x)) = x and E(L(y)) = y, x ∈ R. Then
(a) L is represented by power series on (0, 1). (b) L is represented by power series on
(0,∞). (c) L is not represented by power series. (d) None of the above.
(31) Let coshx =
E(x) + E(−x)
2
and sinhx
E(x)− E(−x)
2
, x ∈ R. Then the following identity
is not true.
(a) sinh(−x) = − sinhx, cosh(−x) = − coshx (b) sinh(x + y) = sinhx cosh(y) +
coshx sinh y (c)
d
dx
sinhx = coshx (d) coshx + sinh2 x = 1
Topology of Metric Spaces and Real Analysis: Practical 3.6
Descriptive Questions 3.6
(1) Show that
∞∑
n=1
xn(1−x) converges uniformly to x on [0, a], where 0 ≤ a < 1, but
∞∑
n=1
xn(1−x)
is not uniformly convergent on [0, 1).
(2) Show that the series
∞∑
n=1
(−x)n(1− x) converges uniformly on [0, 1].
(3) (i) Show that
∞∑
n=1
1
(nx)2
does not converge uniformly on R \ {0} but converges uniformly
on [a,∞), a > 0.
25
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(ii) Show that
∞∑
n=1
1
x2 + n2
is uniformly convergent on R.
(iii) Show that
∞∑
n=1
1
xn + 1
is uniformly convergent on [a,∞), a > 1.
(iv) Show that
∞∑
n=1
xn
xn + 1
is uniformly convergent on [0, a], a < 1 but not pointwise con-
vergent on [1,∞).
(v) Show that
∞∑
n=1
x2
(1 + x2)n
does not converge uniformly on (0,∞) but converges uniformly
on [a,∞), a > 0.
(4) Show that each of the following series of functions converges uniformly on the indicated
interval.
(i)
∞∑
n=1
e−nxxn, [0, A], A > 0.
(ii)
∞∑
n=1
e−nx
n
, x ∈ [a,∞), a > 0.
(iii)
∞∑
n=1
e−nx on [a,∞), a > 0.
(5) If
∞∑
n=1
|an| <∞, then the series
∞∑
n=1
an cosnx and
∞∑
n=1
an sinnx converge on R.
(6) Show that the series
∞∑
n=1
(−1)n(x2 + n)
n2
converges uniformly every bounded subset of R.
(7) Show that
∞∑
n=1
sinnx
np
, p ≤ 1 is uniformly convergent on S = [−pi,−a] ∪ [a, pi] , a > 0.
(8) (i)
∞∑
n=1
2x
e−x2n2
n2
− e
−x2
(n+1)2
(n+ 1)2
 in [a, b]. Show that the series converges uniformly to 2xe−x2
on [a, b]. Hence show that
∞∑
n=1
∫ b
a
2x
e−x2n2
n2
− e
−x2
(n+1)2
(n+ 1)2
 dx = e−a − e−b.
26
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(ii)
∞∑
n=1
xn(1−2xn). Show that the series does not converge pointwise at x = 1 but converges
pointwise to
x
1 + x
on [0, 1). Show that
∫ 1
0
x
1 + x
dx 6=
∞∑
n=1
∫ 1
0
xn(1 − 2xn)dx. Hence
show that the series does not converge uniformly on [0, 1). State the result you used .
(Let D be a bounded subset of R and let f : D −→ R be a function. We say that f is
integrable over D if f is a bounded function and if there are a, b ∈ R with D ⊆ [a, b]
such that the function f ∗ : [a, b] −→ R defined by
f ∗(x) =
{
f(x) if x ∈ D
0 otherwise
is integrable on [a, b]. In this case, the Riemann integral of f over D is defined by∫
D
f(x)dx =
∫ b
a
f ∗(x)dx.
Reference: A Course in Calculus an Real Ananlysis, Sudhir R. Ghorpade, Balmohan
V. Limaye,Second Edition, Springer, pg. no. 216 )
(iii)
∞∑
n=1
[
nx
1 + n2x2
− (n− 1)x
1 + (n− 1)2x2
]
in [0, 1].
Show that
∫ 1
0
[ ∞∑
n=1
[
nx
1 + n2x2
− (n− 1)x
1 + (n− 1)2x2
]]
dx =
∞∑
n=1
∫ 1
0
[
nx
1 + n2x2
− (n− 1)x
1 + (n− 1)2x2
]
dx.
but
∞∑
n=1
[
nx
1 + n2x2
− (n− 1)x
1 + (n− 1)2x2
]
does not converge uniformly on [0, 1].
(9) Show that
∞∑
n=1
1
n3 + n+ x2
is uniformly convergent on R and check that it can be differen-
tiated term by term.
(10) Find the radius of convergence of each of the following power series.
(i)
∞∑
n=0
n3xn
(ii)
∞∑
n=0
2n
n!
xn
(iii)
∞∑
n=0
n3
3n
xn
(iv)
∞∑
n=0
(n3−5n2 +7n−2)xn
(v)
∞∑
n=0
en
n+ 1
xn
(vi)
∞∑
n=0
xn
(n+ 1)
√
n
(11) Find the interval of convergence of the following power series.
(i)
∞∑
n=0
(x− 1)n−1
3nn2
(ii)
∞∑
n=0
n!(x− 2)n
nn
(iii)
∞∑
n=0
(x2 − 1)n
2n
27
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(iv)
∞∑
n=0
(3x+ 6)n
n!
(v)
∞∑
n=0
(x+ 3)n−1
n
(12) Find the radius of convergence of the power series
∞∑
n=0
cnx
n, where cn =
h(h− 1) · · · (h− n+ 1)
n!
.
(13) Consider the power series
∞∑
n=0
cnx
n with integer coefficients. If cn 6= 0 for infinitely many
n, then show that its radius of convergence is at most 1.
(14) Give an example of a power series with radius of convergence = 5 and interval of conver-
gence = (3, 13).
(15) If
∞∑
n=0
cnx
n is a power series such that 0 < α < |cn| < β ∀ n ∈ N where α, β ∈ R, find its
radius of convergence.
(16) Let
∞∑
n=0
anx
n and
∞∑
n=0
bnx
n be power series such that
an =
{
1 if n is square of an integer
0 otherwise
bn =
{
1 if n = k! for some k ∈ N
0 otherwise
.
Find the radius of convergence of
∞∑
n=0
anx
n and
∞∑
n=1
bnx
n.
(17) If 0 < |α| < |β| then find the radius of convergence of
∞∑
n=0
(αn + βn)xn and
∞∑
n=0
(αn + βn)xn.
(18) Show that
1
1− x =
∞∑
n=0
xn for |x| < 1.
(19) By differentiating a suitable power series term by term, obtain the formula,
1 + 2x+ 3x2 + · · ·+ nxn−1 + · · · = 1
(1− x)2
for −a ≤ x ≤ a. What should be the value of ′a′ so that term by term differentiation is
valid?
28
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(20) If sinx = x − x
3
3!
+
x5
5!
+ · · · for x ∈ R and d
dx
(sinx) = cosx, ∀x ∈ R, then show that
cosx = 1− x
2
2!
+
x4
4!
+ · · ·
(21) Show by integrating the series for
1
1 + x
, that log(1 + x) =
∞∑
n=0
(−1)n+1x
n
n
.
(22) By integrating a suitable powe series over an interal [0, t], where 0 ≤ t ≤ 1, show that
1
2
=
∞∑
n=1
1
n!(n+ 2)
.
(23) For |x| < 1, show that sin−1 x =
∞∑
n=0
1.3.5. · · · .(2n− 1)x2n+1
2.4. · · · .(2n)(2n+ 1) .
(24) For |x| < 1, show that tan−1 x =
∞∑
n=0
(−1)nx2n+1
(2n+ 1)
.
(25) Find a series expansion for
∫ x
0
e−t
2
dt for x ∈ R.
(26) If
∞∑
n=0
|an| <∞, prove that
∫ 1
0
( ∞∑
n=0
anx
n
)
dx =
∞∑
n=0
an
n+ 1
.
29
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Topology of Metric Spaces and Real Analysis: Practical 3.7
Miscellaneous.
. Revised Syllabus 2018-19
UNIT I : Continuous functions on Metric Spaces
(1) Let (X, d) and (Y, d′) be metric spaces. Show that f : X → Y is continuous at p ∈ X if
and only if for each sequence (xn) in X converging to p, the sequence (f(xn)) converges to
f(p) in Y .
(2) Let (X, d) and (Y, d′) be metric spaces and f : X −→ Y . Show that the following statements
are equivalent.
(i) f is continuous on X.
(ii) For each open subset G of Y, f−1(G) is an open subset of X.
(iii) For each closed subset F of Y, f−1(F ) is a closed subset of X.
(3) Let (X, d) and (Y, d′) be metric spaces. Show that f : X −→ Y is continuous at p ∈ X if
and only if for each sequence (xn) in X converging to p, the sequence (f(xn)) converges to
f(p) in Y .
(4) Let (X, d) and (Y, d′) be metric spaces. Show that f is continuous at x ∈ X if and only if
for each open subset U of Y containing f(x),∃ an open subset V of X containing x such
that f(V ) ⊆ U .
(5) Let (X, d) and (Y, d′), (Z, d′′) be metric spaces. If f : X −→ Y is continuous and g : Y −→ Z
is continuous, then show using − δ definition or sequential criterion that g ◦ f : X −→ Z
is continuous. Give an example to show that converse of the above statement is not true.
(6) Let (X, d) and (Y, d′) be metric spaces. Show that f : X −→ Y is continuous on X if and
only if for each subset A of X, f(A) ⊆ (f(A)).
(7) Let (X, d) and (Y, d′) be metric spaces. Show that f : XßY is continuous on X if and only
if for each subset B of Y, (f−1(B)) ⊆ f−1(B).
(8) Let (X, d) and (Y, d′) be metric space and f : X −→ R (usual distance) be a continuous
function. If f(x0) > 0 for some x0 ∈ X, show that f(x) > 0,∀x ∈ B(x0, δ).
(9) Let (X, d) and (Y, d′) be metric spaces. When is f : X −→ Y said to be uniformly contin-
uous? Give an example to show that a continuous map need not be uniformly continuous.
(10) Let (X, d) and (Y, d′) be metric spaces. If f, g : X −→ Y are continuous functions, then
show that F = {x ∈ X : f(x) = g(x)} is a closed subset of X. Hence, deduce that if
f(x) = g(x),∀x ∈ D, where D is a dense subset of X, then f = g.
30
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(11) Let (X, d) and (Y, d′) be metric spaces. Show that f : (X, d) −→ (Y, d′) is a continuous
function if and only if f−1(B◦) ⊆ (f−1(B))◦ for each subset B of Y .
(12) Let (X, d) be a metric space and A ⊆ X. Using − δ definition show that fA(x) = d(x,A)
is a continuous map from (X, d) to (R, d) where d is the usual distance on R.
(13) Let f : R −→ R be a continuous function (distance is Euclidean ) and F be a closed
subset of R. Show that A = {x ∈ F : f(x) = 0} is a closed set in R. Is the result true if F
is not closed?
(14) Let (X, d) be a metric space. Show that f : (X, d) −→ (R, d) (where d is usual distance) is
continuous if and only if f−1(−∞, a) and f−1(a,∞) are both open in (X, d) for each a ∈ R.
(15) Show that the metrics d and d1 on a set X are equivalent if and only if i : (X, d) −→ (X, d1)
and i : (X, d1) −→ (X, d) are continuous functions, where i denotes the identity map on X.
(16) Let f : R −→ R (with respect to usual distance) and A = {(x, y) : y < f(x)}, B = {(x, y) :
y > f(x)}. Show that f is continuous on R if and only if A,B are open subsets of (R2, d)
where d is the Euclidean distance.
(17) Let X be a finite set and d be any metric on X. Show that any function f : X −→ Y is
continuous, where (Y, d′) is a metric space.
(18) Let (X, d) be a discrete metric space and (Y, d′) be any metric space. Show that any
function f : X −→ Y is continuous.
(19) Show that any function f : (N, d) −→ (X, d′) is continuous, where d is usual distance on
N and (X, d′) is any metric space.
(20) Show that any function f : (Z, d) −→ (X, d′) is continuous, where d is usual distance on
Z and (X, d′) is any metric space.
(21) (X, d) and (Y, d′) are metric space and f : X −→ Y is continuous. Give examples to show
that
(i) G is an open subset of X does not imply f(G) is an open subset of Y .
(ii) F is a closed subset of X does not imply f(F ) is a closed subset of Y .
(iii) (xn) is a Cauchy sequence in X does not imply the sequence (f(xn)) is a Cauchy in
Y .
(22) Let (X, d) be a metric space and (Y, d′) be any metric spaces. If f : (X, d) −→ (Y, d′) is a
continuous function, then show that f(X) is a compact set.
(23) Let (X, d) and (Y, d′) be metric spaces and f : X −→ Y be continuous. If (X, d) is a
compact metric space, then show that f : X −→ Y is uniformly continuous.
31
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(24) Let (X, d) be a complete metric space. T : X −→ X be a contraction map. Then show
that T has a fixed point.
(25) Let (X, d) be a complete metric space and T : X −→ X be a mapping such that Tm =
T ◦ T ◦ T ◦ . . . ◦ T (mtimes) is a contraction for some fixed m then show T has an unique
fixed point.
(26) Let (X, d) be a compact metric space and T : X −→ X be such that d(T (x), T (y)) <
d(x, y) then show that T has unique fixed point in X.
UNIT II : Connected sets
(1) Let (X, d) be a metric space. Prove that the following statements are equivalent:
(i) X can be expressed as a union of two non-empty separated sets.
(ii) X can be expressed as a union of two non-empty disjoint closed sets.
(iii) X can be expressed as a union of two non-empty disjoint open sets.
(iv) There is a non-empty proper subset of X which is both open and closed.
(2) Show that A is a connected subset of R with respect to the usual distance if and only if it
is an interval.
(3) Let (X, d) be a connected metric space and (Y, d′) be any metric space. If f : (X, d) −→
(Y, d′) is a continuous function, then show f(X) is a connected set.
(4) Show that a metric space (X, d) is connected if and only if every continuous function
f : X −→ {1,−1} is constant.
(5) If a metric space (X, d) is connected and A is a non-empty proper subset of X, then show
that δA, boundary of A is non-empty.
(6) Show that a metric space (X, d) is connected if and only if for each a, b ∈ X, there is a
connected subset E of X such that a, b ∈ E.
(7) Let (X, d) be a metric space. If A is a connected subset of X, and A ⊆ B ⊆ A then show
that B is connected. Hence, show that A is connected. Give an example to show that if
A,C are connected subset of X and A ⊆ B ⊆ C then B need not be connected.
(8) If A and B are connected subset of a metric space (X, d), and A ∩ B 6= ∅, then show that
A ∪B is connected. Give an example to show that A ∩B need not be connected.
(9) Let (X, d) be a metric space. If {Aα : α ∈ Λ} is a family of connected subsets of X such
that ∩α∈ΛAα 6= ∅, then show that ∪α∈ΛAα is connected.
(10) Let (X, d) be a metric space. If {An : n ∈ N} is a family of connected subsets of X such
that An ∩ An+1 6= ∅ for each n ∈ N, then show that ∪n∈NAn is connected.
32
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(11) Prove that an open ball in Rn is a convex set. (The distance being Euclidean). Hence,
deduce that it is path connected.
(12) Show that a path connected subset of Rn is connected.
(13) Let A and B be path connected subsets of a metric space (X, d) such that A ∩ B 6= ∅.
Show that A ∪B is path connected.
(14) Let (X, d) and (Y, d′) be metric spaces. If (X, d) is path connected and f : X −→ Y is
continuous, show that f(X) is path connected.
(15) Let (X, d) be a metric space and A be a non-empty subset of X.
Prove or disprove: IfA is connected, then A◦, and ∂A are connected. Give an example to
show that A◦ and ∂A may be connected, but A may not be connected.
(16) Let (X, d) be a metric space. If A is connected subset of X, then show that A is connected.
Give an example to show that A◦ may not be connected.
UNIT III : Sequences and series of functions
(1) Mn Test: A sequence {fn} of real valued functions on S (S ⊆ R) converges uniformly to
a function f : S −→ R on S if and only if, lim
n−→∞
Mn = 0 where
Mn = sup{|fn(x) − f(x)| : x ∈ S}. Hence show that if there is a sequence (tn) in R such
that |fn(x)−f(x)| ≤ tn for all n ≥ n0 for some n0 ∈ N and for all x ∈ S such that tn −→ 0,
then fn −→ f uniformly on S.
(2) State and prove Cauchy Criterion for uniform convergence of sequences of functions.
(3) Let {fn} be a sequence of real valued functions defined on a set S ⊆ R such that fn −→ f
uniformly on S. If each fn is bounded on S, then prove the following.
(i) f is bounded on S.
(ii) there exists α ∈ R+ such that |fn(x)| ≤ α for all n ∈ N and for all x ∈ S.
(iii) sup{fn(x) : x ∈ S} −→ sup{f(x) : x ∈ S}.
(iv) inf{fn(x) : x ∈ S} −→ inf{f(x) : x ∈ S}.
(4) Let {fn} be a sequence of real valued continuous functions defined on a subset S of R such
that fn −→ f uniformly on S. Then prove the following.
(i) f is continuous on S.
(ii) For any p ∈ S, lim
n−→∞
lim
x−→p
fn(x) = lim
x−→p
lim
n−→∞
fn(x).
33
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(5) Let {fn} be a sequence of real valued R−integrable functions defined on [a, b] such that
fn −→ f uniformly on [a, b] . Then prove that f isR− integrable on [a, b] and lim
n−→∞
∫ b
a
fn(t) dt =∫ b
a
lim
n−→∞
fn(t) dt.
(6) {fn} is a sequence of real valued R−integrable functions on [a, b] converging uniformly to
f on [a, b]. If Fn(x) =
∫ x
a
fn(t) dt then prove that {Fn} converges uniformly to F on [a, b]
where F (x) =
∫ x
a
f(t)dt.
(7) Let {fn} and {gn} be sequences of real valued bounded functions on S subset of R. If {fn}
and {gn} converge uniformly to f and g respectively on S, then prove that {fn ∗ gn} is
uniformly convergent on S.
(8) Let {fn} be a sequence of real valued continuously differentiable functions on [a, b], a < b
such that {fn(x0)} is convergent for some x0 ∈ [a, b] and {f ′n} converges uniformly on [a, b].
Then
(i) there is a continuously differentiable function f on [a, b] such that fn −→ f uniformly
on [a, b] and
(ii) f ′n −→ f ′ uniformly on [a, b].
(9) Let {fn} be a sequence of differentiable real valued functions on a bounded interval I. If
{fn(x0)} is convergent for some x0 ∈ I and {f ′n} converges uniformly to g on I then {fn}
converges uniformly on I and if {fn} converges uniformly to f on I then f is differentiable
on I and f ′ = g on I.
(10) State and prove Cauchy Criterion for Uniform Convergence of a Series
∞∑
n=0
fn of real valued
functions on a subset S of R.
(11) State and prove Weierstrass M-Test for the convergence of a series
∞∑
n=1
fn of real valued
functions defined on subset S of R.
(12) Let {fn} be a sequence of real-valued bounded functions on a set S ⊆ R. If the series∞∑
n=1
fn converges uniformly to the sum function f on S then prove that f is also bounded
on S.
34
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
(13) If {fn} is a sequence if real valued continuous functions on S, S ⊆ R such that
∞∑
n=1
fn con-
verges uniformly to f on S, then prove that f is continuous on S, and for p ∈ S,
∞∑
n=1
lim
x−→p
fn(x) =
lim
x−→p
∞∑
n=1
fn(x).
(14) Let
∞∑
n=1
fn be a series of R−integrable functions on [a, b] , converging uniformly to f on
[a, b], then prove that f is R−integrable on [a, b] and
∫ b
a
f(x) dx =
∞∑
n=1
∫ b
a
fn(x) dx.
(15) If {fn} is a sequence of differentiable functions on [a, b] such that each f ′n is continuous
on [a, b] and if
∞∑
n=1
fn converges to f pointwise on [a, b] and
∞∑
n=1
f ′n converges uniformly on
[a, b] then prove that f ′(x) =
∞∑
n=1
f ′n(x) for a ≤ x ≤ b.
NOTE: For Q. No. (12) to (15), the corresponding result about uniform convergence of
sequence of functions can be used directly.
(16) If the power series
∞∑
n=0
cnx
n converges at x1 ∈ R, x1 6= 0 and diverges at x2 ∈ R then the
power series
∞∑
n=0
|cnxn| converges for all x ∈ R with |x| < |x1| and diverges for all x ∈ R
with |x| > |x2|.
(17) A power series
∞∑
n=1
cnx
n is either absolutely convergent for all x ∈ R, or there is a unique
real number r ≥ 0 such that the series is absolutely convergent for each x ∈ R with |x| < r
and is divergent for each x ∈ R with |x| > r.
(18) Let
∞∑
n=0
cnx
n be a power series with coefficients in R. Let α = lim sup
n−→∞
∣∣∣cn∣∣∣ 1n . Then the
radius of convergence r of
∞∑
n=0
cnx
n is
1
α
(if α = 0, r =∞ and if α =∞, r = 0) (Statement
Only).
35
US/AMT603 Sem VI,Paper3:Topology of Metric Spaces and Real Analysis, Rev. Syl. 2018
Definition: limit superior of a sequence
(
lim sup
n−→∞
an
)
: Let (an) be a sequence in R.
i. If (an) is not bounded above then lim sup
n−→∞
an =∞
ii. If (an) is bounded above then for each n ∈ N, define, Mn = sup{ak : k ≥ n}. Then
sequence (Mn) is monotonic decreasing. If sequence (Mn) is bounded below then it is
convergent. In such case, lim sup
n−→∞
an = lim
n−→∞
Mn.
iii. If sequence (Mn) is not bounded below then lim sup
n−→∞
an = −∞.
It can be proved that if sequence (an) is convergent then lim sup
n−→∞
an = lim
n−→∞
an
(19) Let
∞∑
n=0
cnx
n be a power series with coefficients in R and there exist n0 ∈ N such that
cn 6= 0, ∀n ≥ n0. Let α = lim
n−→∞
∣∣∣cn+1
cn
∣∣∣. Then the radius of convergence r of ∞∑
n=0
cnx
n is
1
α
(if α = 0, r =∞ and if α =∞, r = 0) (Statement Only).
(20) Let r be the radius of convergence of a power series
∞∑
n=1
cnx
n. If s ∈ R is such that
0 < s < r, then prove that the power series converges uniformly on [−s, s]. Further, let
f : (−r, r) −→ R be the sum function of the power series
∞∑
n=0
cnx
n then prove that
(i) f is continuous on (−r, r).
(ii) For every x ∈ (−r, r),
∫ x
0
f(t)dt =
∞∑
n=0
cn
xn+1
n+ 1
(iii) f is differentiable on (−r, r) and f ′(x) =
∞∑
n=1
ncnx
n−1 for x ∈ (−r, r).
(iv) f is infinitely differentiable on (−r, r), and cn = f
(n)(0)
n!
for n ∈ N, c0 = f(0).
36