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Examples. Justify your answers briefly. (a) There is a function f : R → R so that f(-1) = -1 and f(1) = 1, but so that f is never 0. What else can you say about f? (b) Explain briefly why {x € Q20 : x < 5}|{x € Q2° : x > 5} is a Dedekind cut. What positive real number does it represent? (c) Give an example of a sequence a, whose image set {a, : n e N} is not compact. (d) Give an example of a subset of R? that is neither open nor closed.

Examples. Justify your answers briefly. (a) There is a function f : R → R so that f(-1) = -1 and f(1) = 1, but so that f is never 0. What else can you say about f? (b) Explain briefly why {x € Q20 : x < 5}|{x € Q2° : x > 5} is a Dedekind cut. What positive real number does it represent? (c) Give an example of a sequence a, whose image set {a, : n e N} is not compact. (d) Give an example of a subset of R? that is neither open nor closed.

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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Examples. Justify your answers briefly. (a) There is a function f : R → R so that f(-1) = -1 and f(1) = 1, but so that f is never (0. What else can you say about f? (b) Explain briefly why {x € Q20 : x < 5}|{x € Q2° : x > 5} is a Dedekind cut. What positive real number does it represent? (c) Give an example of a sequence a, whose image set {a, : n e N} is not compact. (d) Give an example of a subset of R? that is neither open nor closed.
Transcribed Image Text:Examples. Justify your answers briefly. (a) There is a function f : R → R so that f(-1) = -1 and f(1) = 1, but so that f is never (0. What else can you say about f? (b) Explain briefly why {x € Q20 : x < 5}|{x € Q2° : x > 5} is a Dedekind cut. What positive real number does it represent? (c) Give an example of a sequence a, whose image set {a, : n e N} is not compact. (d) Give an example of a subset of R? that is neither open nor closed.
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