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Math Help - Compact continuity question.

  1. #1
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    Compact continuity question.

    Prove that if F\subset\mathbb{R}, is not compact. Then there exists a continuous function f:F\rightarrow\mathbb{R} which isn't bounded on F

    Any help would be appreciated.
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  2. #2
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    Quote Originally Posted by skamoni View Post
    Prove that if F\subset\mathbb{R}, is not compact. Then there exists a continuous function f:F\rightarrow\mathbb{R} which isn't bounded on F
    If F is not compact then it is either unbounded or not closed (or both). If it is unbounded then (obviously) the function f(x) = x is unbounded on F. If F is not closed, let z be a point in the closure of F but not in F, and let f(x) = 1/(xz).
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  3. #3
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    Quote Originally Posted by skamoni View Post
    Prove that if F\subset\mathbb{R}, is not compact. Then there exists a continuous function f:F\rightarrow\mathbb{R} which isn't bounded on F

    Any help would be appreciated.
    Two possibilities: (i) either F is not closed, (ii) F is not bounded.

    If F is not closed then there exists p\in \partial F so that p\not \in F. Now define f(x) = \tfrac{1}{|p-x|}.
    This function is continous and unbounded since x gets arbitrary close to p.

    If F is not bounded then for some p\in F define f(x) = |p-x|.
    Certainly this is not bounded since x gets arbitrary away from p.

    EDIT: Opalg beat me too it .

    By the way I think it is possible to find a \mathcal{C}^{\infty} function , do you agree ?
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