In Quantum Physics Gasiorowicz states:
Incidentally, had we allowed for discontinuities in $\psi$(x, t) we would have been led to delta functions in the flux, and hence in the probability density, which is unacceptable in a physically observed quantity.
The main concern over here is that the probability density can't be a delta function, but why? If we have $P=\delta(x)$, wouldn't it represent a particle that is localised at $x=0$, and has no spatial extent? If so, then what is the issue?
