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
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ISBN:9780470458365
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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.
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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