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Continuity, Closed & Open Sets and Differentiability

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Prove OR disprove the following statements. Explain.

(i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s).

(ii) There is a nonempty set S in R such that S is neither open nor closed.

(iii) There is a nonempty set S in R such that S is both open and closed.

(iv) Let a < b. if f: (a,b] -> R is continuous, then f is uniformly continuous on (a,b].

(v) Let a < b, and let f: [a,b] -> R be continuous on [a,b] and differentiable on (a,b). If f(a) < f(b), then there exists x in (a,b) such that f'(x) > 0.

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Solution Summary

Continuity, Closed and Open Sets and Differentiability are investigated in the following problems.

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(i) There is a nonempty set S in R ( real numbers) such that S contains none of its limit-point(s).

Yes
Let S={1,1/2,1/3,...,1/n,...}, then the limit point of S is 0, but S does not contain 0.

(ii) There is a nonempty set S in R such that S is neither open nor closed.

Yes
Let S=(0,1]. S is not closed because 0 is a limit point of S, but ...

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