Single beats, infinitely many frequencies

[metroplex021]

Does anyone know how to prove that if we want to make a single wavepacket in space we necessarily have to superpose a continuous infinity of frequencies? Any help much appreciated!

 

[Oxvillian]

A wavepacket is an integrable function, and the Fourier transform of an integrable function is continuous.

 

[metroplex021]

Thanks a lot, that helps!