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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 Preview

    (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 ...

    Solution Summary

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

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