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Does anyone knows how to prove that the function $f$ defined from $[0,1] \rightarrow \mathbb{R}$ by $\sin(\frac{1}{2\sqrt{x}})$ if $x\ne 0$ and equal $0$ if $x=0$ is Riemann integrable? Thanks.

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  • $\begingroup$ Just say it's bounded near 0 and continuous elsewhere. $\endgroup$ – user2345215 Feb 28 '14 at 2:16
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HINT: If $f$ is bounded on $[0,1]$ and integrable on $[c,1]$ for every $0<c<1$, is $f$ integrable on $[0,1]$?

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