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Math Help - counter examples with real functions

  1. #1
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    counter examples with real 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
    D) if D is not closed, then f(D) is not closed
    E) if D is not compact, then f(D) is not compact
    F) if D is unbounded, then f(D) is unbounded
    G) if D is finite, then f(D) is finite
    H) if D is infinite, then f(D is infinite
    I) if D is an interval, then f(D) is an interval
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  2. #2
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    Quote Originally Posted by slowcurv99 View Post
    Let f: D->R be continuous. For each of the following, prove or give a counter example.
    Say D is not empty.
    A) if D is open, then f(D) is open
    Consider f(x)=1
    B) if D is closed, then f(D) is closed
    Consider f(x)=1
    D) if D is not closed, then f(D) is not closed
    Consider f(x)=x^2 on (0,1).
    E) if D is not compact, then f(D) is not compact
    Heine-Borel Theorem: Compact = Closed + Bounded.
    We already talked about closed getting mapped to closed.
    You can answer this now.
    F) if D is unbounded, then f(D) is unbounded
    Consider D=(-oo,+oo) and f(x)=0.
    G) if D is finite, then f(D) is finite
    Consider f(x)=1/x on (0,1)
    H) if D is infinite, then f(D) is infinite
    How is that different from unbounded?
    I) if D is an interval, then f(D) is an interval
    It is either an interval or a single point!
    Last edited by ThePerfectHacker; April 17th 2007 at 06:21 PM.
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