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# Continuous bounded function which doesn't attain either bound watch

1. I'm trying to find a continuous bounded function f:[0,1) -> R which doesn't attain either bound.

Is this true for f(x) = x*sin(1/x)?
I know it is continuous and bounded and think I can show this, but I'm not 100% about the idea that it doesn't obtain either bound.
2. (Original post by Will.Honeyman)
I'm trying to find a continuous bounded function f:[0,1) -> R which doesn't attain either bound.

Is this true for f(x) = x*sin(1/x)?
I know it is continuous and bounded and think I can show this, but I'm not 100% about the idea that it doesn't obtain either bound.
Seems to hit its lower bound. See attached.

Try moving the "nasty stuff" so that it's towards the open end of the interval, rather than the closed end, and "play with it a bit"
4. Looks good.

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Updated: January 31, 2010
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