Continuous functions

**TriAngle** · February 1st 2009, 07:45 AM

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?

**flyingsquirrel** · February 1st 2009, 09:48 AM

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 $\left|x\sin\left(\tfrac{1}{1-x}\right)\right|\leq \left|x\right| \times 1<1$ .

**TriAngle** · February 1st 2009, 10:48 AM

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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