Question 9. (a) Given a continuous function f: RR and a connected subset S € R, is f¹(S) connected? Justify your answer.
Question 9. (a) Given a continuous function f: RR and a connected subset S € R, is f¹(S) connected? Justify your answer.
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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![Question 9.
(a) Given a continuous function f: RR and a connected subset SE R, is f¹(S)
connected? Justify your answer.
(b) Given two metric spaces X,p>, <Y,T> and a function f : X → Y that is uniformly
continuous on SC X. If a sequence (Tn)neN € S is Cauchy in X, show that (f(n))neN
is Cauchy in Y.
(c) Given two sequences (fn)neN, (9n)neN C C[0, 1] of continuous functions on the closed unit
interval [0, 1] defined by
nx
nx
fn(r)
and g(x)
1+nx²¹
1+n²x².
Find the limit f and g, respectively of each sequence, if it exists. Which of these sequences
converge uniformly on [0, 1]? That is, do f and g belong to C[0, 1] or not?
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Transcribed Image Text:Question 9.
(a) Given a continuous function f: RR and a connected subset SE R, is f¹(S)
connected? Justify your answer.
(b) Given two metric spaces X,p>, <Y,T> and a function f : X → Y that is uniformly
continuous on SC X. If a sequence (Tn)neN € S is Cauchy in X, show that (f(n))neN
is Cauchy in Y.
(c) Given two sequences (fn)neN, (9n)neN C C[0, 1] of continuous functions on the closed unit
interval [0, 1] defined by
nx
nx
fn(r)
and g(x)
1+nx²¹
1+n²x².
Find the limit f and g, respectively of each sequence, if it exists. Which of these sequences
converge uniformly on [0, 1]? That is, do f and g belong to C[0, 1] or not?
=
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