Is the function $f(x)=\sin(1/x)$ Lebesgue integrable on $(0,1]$?
I know that, as $f$ is continuous on the set, it is a measurable function. However, I'm stumped on how to go on. A nudge in the right direction would be greatly appreciated.
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It is even Riemann integrable. – Marc van Leeuwen Feb 10 '15 at 11:17
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Does Riemann imply Lebesgue on finite intervals then? – Ori Feb 10 '15 at 11:19
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1Yes – Marc van Leeuwen Feb 10 '15 at 11:27
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Hint: $|f(x)|\le1$ for all $x\in(0,1]$, so $f$ is uniformly bounded.
Jason
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1What does it mean for a single function to be uniformly bounded? – Marc van Leeuwen Feb 10 '15 at 11:20
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