If f : [0, 1] → R is continuous and (In)neN is a sequence of functions that converge pointwise to some function 9, and if hn(x) = f(gn(x)) for all x ER, then the sequence (hn)neN converges pointwise. a. True, by sequential characterisation of the continuity of f. b. False, because f(s) could go to infinity as s goes to +∞ or -∞. c. False, because some gn could be discontinuous. d. True, because by preservation of compact sets f([0, 1]) is compact, and all (hn(x))neN belong to that compact set.
If f : [0, 1] → R is continuous and (In)neN is a sequence of functions that converge pointwise to some function 9, and if hn(x) = f(gn(x)) for all x ER, then the sequence (hn)neN converges pointwise. a. True, by sequential characterisation of the continuity of f. b. False, because f(s) could go to infinity as s goes to +∞ or -∞. c. False, because some gn could be discontinuous. d. True, because by preservation of compact sets f([0, 1]) is compact, and all (hn(x))neN belong to that compact set.
Functions and Change: A Modeling Approach to College Algebra (MindTap Course List)
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Author:Bruce Crauder, Benny Evans, Alan Noell
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Chapter1: Functions
Section1.2: Functions Given By Tables
Problem 32SBE: Does a Limiting Value Occur? A rocket ship is flying away from Earth at a constant velocity, and it...
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![Q4 MULTIPLE CHOICE One answer only
If ƒ : [0,1] → R is continuous and (gn)neN is a sequence of functions that converge pointwise to some
function g, and if hn(x) = f(gn(x)) for all à € R, then the sequence (hn)neN converges pointwise.
a. True, by sequential characterisation of the continuity of ƒ.
b. False, because f(s) could go to infinity as s goes to +∞ or —∞.
c. False, because some gn could be discontinuous.
d. True, because by preservation of compact sets f([0, 1]) is compact, and all (hn (x))neN belong to that
compact set.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3cb672f7-47ed-4ee3-be4e-71db737c6150%2F08f863a0-ed52-47a2-8c60-c1040fba56e6%2Fb7hnh8a_processed.png&w=3840&q=75)
Transcribed Image Text:Q4 MULTIPLE CHOICE One answer only
If ƒ : [0,1] → R is continuous and (gn)neN is a sequence of functions that converge pointwise to some
function g, and if hn(x) = f(gn(x)) for all à € R, then the sequence (hn)neN converges pointwise.
a. True, by sequential characterisation of the continuity of ƒ.
b. False, because f(s) could go to infinity as s goes to +∞ or —∞.
c. False, because some gn could be discontinuous.
d. True, because by preservation of compact sets f([0, 1]) is compact, and all (hn (x))neN belong to that
compact set.
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