6. Let f,g: R→ R. be continuous functions and define S = {r ER: f(x) = g(x) = Z}, where Z stands for the set of all integers. Let (n) be a convergent sequence in S and let z = limn→∞ n. Can we say that ES? Would your answer change if we replace Z with the set Q of all rational numbers? Justify your answer.

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Question 6 in the attached image.

6. Let f,g: R –→R be continuous functions and define
S = {r € R : f(r) – g(x) E Z},
where Z stands for the set of all integers. Let (xn) be a convergent sequence in S and let r =
limno Tn. Can we say that r e S? Would your answer change if we replace Z with the set Q of all
rational numbers? Justify your answer.
Transcribed Image Text:6. Let f,g: R –→R be continuous functions and define S = {r € R : f(r) – g(x) E Z}, where Z stands for the set of all integers. Let (xn) be a convergent sequence in S and let r = limno Tn. Can we say that r e S? Would your answer change if we replace Z with the set Q of all rational numbers? Justify your answer.
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