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Math Help - Open and closed sets and functions

  1. #1
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    Open and closed sets and functions

    Let f->R be continuous. For each of the following, prove or give a counter example:

    a) If D is open, then f(D) is open
    b) if D is closed, then f(D) is closed
    c) If D is unbounded, then f(D) is unbounded
    d) If D is finite, then f(D) is finite
    e) If D is infinite, then f(d) is infinite.

    I need to know counter examples for a,b,c, and e and how to prove d is true.
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  2. #2
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    Quote Originally Posted by algebrapro18 View Post
    a) If D is open, then f(D) is open
    False. Consider constant functions.
    b) if D is closed, then f(D) is closed
    Consider f(x) = e^{-x} and D = \mathbb{R}.
    c) If D is unbounded, then f(D) is unbounded
    Consider constant function.
    d) If D is finite, then f(D) is finite
    The image of a finite set is always finite.
    e) If D is infinite, then f(d) is infinite.
    Consider constant functions.
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