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$f:[0,\infty)\longrightarrow[0,\infty)$ continuous, non negative.

prove or give a counter example

1) if ${\displaystyle \intop_{1}^{\infty}f\left(x\right)dx}$ exists then f is bounded.

2) if ${\displaystyle \intop_{1}^{\infty}f\left(x\right)dx}$ exists then ${\displaystyle \intop_{1}^{\infty}f^{2}\left(x\right)dx}$ exists as well.

I think both claims are wrong, but all the counter examples I could think of didn't work out.

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Try the following: $f$ is a function which is $n$ at $n$, $0$ at $n\pm1/n^3$, and piece wise linear on $[n,n+1/n^3]$ and $[n-1/n^3,n]$.

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    @sonyjimbo, Sketch out the function, it looks like a sequence of triangular spikes (and zero everywhere else) with heights $n$ and widths $2/n^3$ so you get area contributions of $1/n^2$ which converges. Now for the second part, underestimate $\int f^2 $ as follows: In the middle section of each spike, which has width $1/n^3$, the height of the spike is greater than $n/2.$ So then square the function, and the area contribution from this same interval is greater than $(n^2/4)*(1/n^3) = 1/(4n)$, which will diverge.2012-05-26