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Math Help - does such a function exist?

  1. #1
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    does such a function exist?

    I have a function h with the property
    <br />
\forall x,y \in R,\ x\leq y \rightarrow h\left(x\right) \leq h\left(y\right)<br />
    Now A is a nonempty set which has a supremum
    Are there functions such that:

    <br />
1. h\left(A\right) \ has \ not\ a\ supremum

    <br />
2. h^{-1}\left(A\right) \ has\ not\ a\ supremum<br /> <br />
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  2. #2
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    For the first question, let h(x)=-1/x (-1<x<0) .
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  3. #3
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    I think, for the second question, the answer is NO.
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  4. #4
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    but -1/x is not monotonic on the whole real line
    for instance  -1<2 \ and \  -1/-1 = 1>-1/2

    I think that the second is true for for instance h^{-1}=-x

    is this right?
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  5. #5
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    Quote Originally Posted by hanns View Post
    but -1/x is not monotonic on the whole real line
    for instance  -1<2 \ and \  -1/-1 = 1>-1/2
    Then the answer is No.
    If A has a supremum,(say a), but h(A) doesn't, then what is h(a+1)?
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