Let ƒ : R¹ → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f-¹(X) CR¹ is closed. (ii) For every closed set X CR", the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f−¹(K) C R” is se- quentially compact. (iv) For every sequentially compact set K CR¹, the set f(K) C Rk is sequen- tially compact.
Let ƒ : R¹ → Rk be a continuous function. Which of the following claims are true? Justify your answers give a counterexample (i) For every closed set X CR, the set f-¹(X) CR¹ is closed. (ii) For every closed set X CR", the set f(X) C Rk is closed. (iii) For every sequentially compact set KC Rk, the set f−¹(K) C R” is se- quentially compact. (iv) For every sequentially compact set K CR¹, the set f(K) C Rk is sequen- tially compact.
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter6: Applications Of The Derivative
Section6.CR: Chapter 6 Review
Problem 2CR
Related questions
Question
![Let f: R → Rk be a continuous function. Which of the following
claims are true? Justify your answers
give a counterexample
(i) For every closed set X CR, the set f−¹(X) C Rn is closed.
(ii) For every closed set X CR, the set f(X) C Rk is closed.
(iii) For every sequentially compact set KC Rk, the set f-¹(K) C Rª is se-
quentially compact.
(iv) For every sequentially compact set KCR, the set f(K) C Rk is sequen-
tially compact.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4a97bb2c-6d83-4604-9a70-f559a7370826%2Ffd38f00d-ad4f-474b-9bbe-6bae07c33336%2Fz5jq8h3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Let f: R → Rk be a continuous function. Which of the following
claims are true? Justify your answers
give a counterexample
(i) For every closed set X CR, the set f−¹(X) C Rn is closed.
(ii) For every closed set X CR, the set f(X) C Rk is closed.
(iii) For every sequentially compact set KC Rk, the set f-¹(K) C Rª is se-
quentially compact.
(iv) For every sequentially compact set KCR, the set f(K) C Rk is sequen-
tially compact.
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