Convergence of an integral - book vs. me

Let f(t) be a function in [tex]L^2[/tex]. I am interested under which conditions converges the integral

[tex]\int_0^\infty \frac{|F(\omega)|^2}{\omega} d\omega[/tex]

where F(omega) denotes the Fourier transform of f.

My book, well, several books actually, say the sufficient conditions are
1) [tex]F(0) = 0[/tex] (naturally)
2) F is continuously differentiable ([tex]C^1[/tex])

I don't understand why the differentiability is neccessary. My conjecture - if F is continuous (not neccessarily C1), then the integral converges around zero because of the first condition and thus everywhere since F is in L2 because f was in L2 and the Fourier transform maps L2 onto L2, so there is no problem around infinity.

Where am I wrong? Thanks for any ideas, H.

 

Sorry for the duplicate.