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 $[0,1]$.