a function that is in $L^2$ but not in $L^p$ for $p < 2$

Question

I'm taking the space to be $\mathbb{R}$ equipped with the standard (Lebesgue) measure on it. I need a function which, when squared, yields a convergent integral but when raised to any power between 1 and 2 (2 not inclusive), doesn't. Clearly this function can't have compact support...I can see that much. But I cannot think of an example of such a function.

Answer 1

Note that $\int_2^{\infty}x^{-1}(\log x)^{-2}\,dx$ converges, so $f(x)=x^{-1/2}\log x$ should do (just define it to be zero for $x<2$).