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Math Help - Countinous bounded function that does not attain Its bounds

  1. #1
    LHS
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    Exclamation Countinous bounded function that does not attain Its bounds

    Can anyone think of a continuous function f:[0,1) --> R that is bounded but does not attain one of it's bounds?

    How about either of it's bounds?
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  2. #2
    A Plied Mathematician
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    f(x) = x, domain of [0,1), does not attain its bound of 1 on that interval. As for either of its bounds, you'll have to get way more fancy. How about

    f(x)=e^{x}\sin\left(\dfrac{1}{1-x}\right)?
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  3. #3
    LHS
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    Oh that's good! So as x tends to 1 it becomes undefined, but still oscillates all the way towards zero, and the exponential causes each oscillation to be greater than the last, hence it never attains either of its bounds?
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  4. #4
    A Plied Mathematician
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    That's the idea, anyway.
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