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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- Jan 31st 2011, 03:43 PMLHSCountinous 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? - Jan 31st 2011, 03:54 PMAckbeet
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

$\displaystyle f(x)=e^{x}\sin\left(\dfrac{1}{1-x}\right)?$ - Jan 31st 2011, 04:11 PMLHS
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?

- Jan 31st 2011, 04:26 PMAckbeet
That's the idea, anyway.