4. Let D be a subset of R. (a) A point xo in D is said to be an isolated point of D provided that there is an r > 0 such that the only point of D in the interval (xor, xo + r) is xo itself. Prove that a point xo in D is either an isolated point of D or an limit point of D. (b) Let f Z→ R be a function defined on the set of integers Z. When will f be a continuous function? Find some example that f is continuous or discontinuous. Justify your answer.
4. Let D be a subset of R. (a) A point xo in D is said to be an isolated point of D provided that there is an r > 0 such that the only point of D in the interval (xor, xo + r) is xo itself. Prove that a point xo in D is either an isolated point of D or an limit point of D. (b) Let f Z→ R be a function defined on the set of integers Z. When will f be a continuous function? Find some example that f is continuous or discontinuous. 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
![4. Let D be a subset of R.
(a) A point xo in D is said to be an isolated point of D provided that there is an r > 0
such that the only point of D in the interval (xo - r, xo + r) is xo itself. Prove that a
point xo in D is either an isolated point of D or an limit point of D.
(b) Let f: Z → R be a function defined on the set of integers Z. When will f be a
continuous function? Find some example that f is continuous or discontinuous. Justify
your answer.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80f0fd83-d44a-4dc1-b23c-0550049a776e%2F517548c3-174b-464b-805d-0eb6cc7ec333%2F824o2cb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4. Let D be a subset of R.
(a) A point xo in D is said to be an isolated point of D provided that there is an r > 0
such that the only point of D in the interval (xo - r, xo + r) is xo itself. Prove that a
point xo in D is either an isolated point of D or an limit point of D.
(b) Let f: Z → R be a function defined on the set of integers Z. When will f be a
continuous function? Find some example that f is continuous or discontinuous. Justify
your answer.
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