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Let $f$ be a function on $[0,1]$ and continuous on $(0,1]$.

I want to find a function $f$ s.t. {$\int_{[1/n,1]}f$} converges and yet $f$ is not $L$-integrable over $[0,1]$.

My attempts:

I've found $f(x)=(1/x)Sin(1/x)$ but I can not prove that $f$is not $L$-integrable over $[0,1]$.

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    Oops! I understand what you mean now by $\{\int_{[\frac1n,1]}f\}$ converges. I have deleted my answer.2012-11-05

3 Answers 3