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Math Help - range of unbounded function

  1. #1
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    range of unbounded function

    The function h:\mathbb{R}\rightarrow\mathbb{R} is continuous on \mathbb{R} and let
    h\left (\mathbb{R}  \right )= \left\{h(x):x\in\mathbb{R}  \right \}
    be the range of h. Prove that if h\left (\mathbb{R}  \right ) is not bounded above and not bounded below, then h\left (\mathbb{R}  \right )=\mathbb{R}.


    I tried using the definitions of not bounded above and not bounded below, i.e.

    not bounded above: \exists{u}\in\mathbb{R}, \forall{s}\in\mathbb{R}\Rightarrow{s}>{u}
    not bounded below: \exists{v}\in\mathbb{R}, \forall{v}\in\mathbb{R}\Rightarrow{s}<{v}

    Am I on the right track? How do I proceed from here?
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  2. #2
    MHF Contributor FernandoRevilla's Avatar
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    Re: range of unbounded function

    If y\in\mathbb{R} , as h(\mathbb{R}) is not bounded then there exist x_1,x_2\in\mathbb{R} such that h(x_1)<y<h(x_2) . Now, apply the Intermediate Value Theorem .
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  3. #3
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    Re: range of unbounded function

    Ok. I got it thanks!
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