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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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 $\displaystyle f(x)=e^{x}\sin\left(\dfrac{1}{1-x}\right)?$
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?
That's the idea, anyway.
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