Continuity and its relationship with asymptotic spectral decay
I suppose it would be sufficient to show that a discontinuous function in the time domain decays as $1/\omega$ in the frequency domain (e.g., $\textrm{rect}\Leftrightarrow\textrm{sinc}$), and then use the fact that each integration in the time domain adds a multiplicative factor of $1/j\omega$ in the frequency domain.