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Thread: Continuous functions

  1. #1
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    Continuous functions

    Does there exist a function f, defined by $\displaystyle
    f:\left[ {0,1} \right) \to \mathbb{R}
    $, 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 $\displaystyle \begin{array}{rl} f:[0,1)&\rightarrow \mathbb{R}\\ x & \mapsto \sin x\end{array}$ is bounded above and below by 2 and -2 respectively and $\displaystyle f$ never attains these two bounds...

    Another example : let $\displaystyle \begin{array}{rl} f:[0,1)&\rightarrow \mathbb{R}\\ x & \mapsto x\sin\left(\frac{1}{1-x}\right)\end{array}$. This function is bounded above and below by $\displaystyle 1$ and $\displaystyle -1$ respectively but $\displaystyle f(x)$ never equals $\displaystyle \pm1$ since $\displaystyle \left|x\sin\left(\tfrac{1}{1-x}\right)\right|\leq \left|x\right| \times 1<1$.
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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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