2) Q2 MULTIPLE CHOICE One answer only. Let g: [0, 1] → R. be a function such that g(0) = 0, and (xn)neN a sequence in (0, 1) that converges to 0. If g(xn) →0 as n→∞ then g is continuous at 0. a. True, because any function defined on the interval [0, 1] is continuous by IVT. b. False, here is a counter-example: g(x) = sin(1/x) (with g(0) = 0) and xn = 1/(nπ). c. True, by sequential characterisation of the continuity at 0. d. False, here is a counter-example: (n)neN = (0, 0, 0, ...). defined by g(0) = 0 and g(x) = 1 if x > 0, and the sequence
2) Q2 MULTIPLE CHOICE One answer only. Let g: [0, 1] → R. be a function such that g(0) = 0, and (xn)neN a sequence in (0, 1) that converges to 0. If g(xn) →0 as n→∞ then g is continuous at 0. a. True, because any function defined on the interval [0, 1] is continuous by IVT. b. False, here is a counter-example: g(x) = sin(1/x) (with g(0) = 0) and xn = 1/(nπ). c. True, by sequential characterisation of the continuity at 0. d. False, here is a counter-example: (n)neN = (0, 0, 0, ...). defined by g(0) = 0 and g(x) = 1 if x > 0, and the sequence
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![(2) Q2MULTIPLE CHOICE One answer only
Let g : [0, 1] → R be a function such that g(0)
g(xn) → 0 as n →∞ then g is continuous at 0.
=
0, and (xn)n≤N a sequence in (0, 1) that converges to 0. If
a. True, because any function defined on the interval [0, 1] is continuous by IVT.
b. False, here is a counter-example: g(x) = sin(1/x) (with g(0) = 0) and în = 1/(nâ).
c. True, by sequential characterisation of the continuity at 0.
d. False, here is a counter-example: g defined by g(0) = 0 and g(x)
(n)neN = (0, 0, 0, ...).
=
1 if x > 0, and the sequence](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe1fc35-672a-49fd-831b-9642c77888ed%2Ffd1b28b6-6b18-4ee2-8b0f-b6bd519c799b%2Ff08buce_processed.png&w=3840&q=75)
Transcribed Image Text:(2) Q2MULTIPLE CHOICE One answer only
Let g : [0, 1] → R be a function such that g(0)
g(xn) → 0 as n →∞ then g is continuous at 0.
=
0, and (xn)n≤N a sequence in (0, 1) that converges to 0. If
a. True, because any function defined on the interval [0, 1] is continuous by IVT.
b. False, here is a counter-example: g(x) = sin(1/x) (with g(0) = 0) and în = 1/(nâ).
c. True, by sequential characterisation of the continuity at 0.
d. False, here is a counter-example: g defined by g(0) = 0 and g(x)
(n)neN = (0, 0, 0, ...).
=
1 if x > 0, and the sequence
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