Why do multiparticle states present continuous in spectral function?

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I have learned the fact from Peskin QFT book,that one-particle state presents a delta function form in spectral function at s=m^2,while multiparticle states present a continuous form begin at s=4m^2,but i don't really understand the reason.What cause the difference between one-particle state and multiparticle states?Could someone explain this a bit clearer from the viewpoint of mathematics or physics or both?
 
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Physically, if we have two particles, we can give them equal and opposite three-momentum (of any magnitude). Then their total three-momentum is zero, but their total energy varies continuously from E=2m (when they each have zero three-momentum) up to infinity. Similarly with 3 or more particles.