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Math Help - Continuous functions

  1. #1
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    Continuous functions

    Does there exist a function f, defined by <br />
f:\left[ {0,1} \right) \to \mathbb{R}<br />
, which is bounded above and below but does not ever attain the upper bound nor the lower bound? Can anyone give an example of such a function?
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hello

    Well \begin{array}{rl} f:[0,1)&\rightarrow \mathbb{R}\\ x & \mapsto \sin x\end{array} is bounded above and below by 2 and -2 respectively and f never attains these two bounds...

    Another example : let \begin{array}{rl} f:[0,1)&\rightarrow \mathbb{R}\\ x & \mapsto x\sin\left(\frac{1}{1-x}\right)\end{array}. This function is bounded above and below by 1 and -1 respectively but f(x) never equals \pm1 since \left|x\sin\left(\tfrac{1}{1-x}\right)\right|\leq \left|x\right| \times 1<1.
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  3. #3
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    That's what I was thinking. It seems like a simple solution should work, but the problem is presented in such a way as to make it seem like it should be harder.
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