Examples. Justify your answers briefly. (a) Describe the Dedekind cut corresponding to √11. Your description should not refer directly to any irrational numbers. (b) Give an example of a closed subset of R that is not sequentially compact. (c) Give an example of a real function f, defined at all points, such that f(0) ) = −1 and f(1) = 3, but where f(x) is never 0. (d) Give an example of a continuous function f : [0, 1] → [0, 1] having exactly one fixed point.
Examples. Justify your answers briefly. (a) Describe the Dedekind cut corresponding to √11. Your description should not refer directly to any irrational numbers. (b) Give an example of a closed subset of R that is not sequentially compact. (c) Give an example of a real function f, defined at all points, such that f(0) ) = −1 and f(1) = 3, but where f(x) is never 0. (d) Give an example of a continuous function f : [0, 1] → [0, 1] having exactly one fixed point.
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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![Examples. Justify your answers briefly.
(a) Describe the Dedekind cut corresponding to √11. Your description should
not refer directly to any irrational numbers.
(b) Give an example of a closed subset of R that is not sequentially compact.
(c) Give an example of a real function f, defined at all points, such that
f(0) = -1 and f(1) = 3, but where f(x) is never 0.
(d) Give an example of a continuous function f : [0, 1] → [0, 1] having exactly
one fixed point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb4074fd-9e3f-45a1-be46-9ca60cf326e6%2Fe17c7931-0871-4473-ae9e-e10bfd9f4701%2Fndjbj3t_processed.png&w=3840&q=75)
Transcribed Image Text:Examples. Justify your answers briefly.
(a) Describe the Dedekind cut corresponding to √11. Your description should
not refer directly to any irrational numbers.
(b) Give an example of a closed subset of R that is not sequentially compact.
(c) Give an example of a real function f, defined at all points, such that
f(0) = -1 and f(1) = 3, but where f(x) is never 0.
(d) Give an example of a continuous function f : [0, 1] → [0, 1] having exactly
one fixed point.
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