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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.

Advanced Engineering Mathematics
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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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.
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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