Differentiability of Audio Signals
Since the length of the integral is finite, there is no possibility that it can ``blow up'' due to the weighting by in the frequency domain introduced by differentiation in the time domain.
A basic Fourier property of signals and their spectra is that a signal cannot be both time limited and frequency limited. Therefore, by conceptually ``lowpass filtering'' every audio signal to reject all frequencies above kHz, we implicitly make every audio signal last forever! Another way of saying this is that the ``ideal lowpass filter `rings' forever''. Such fine points do not concern us in practice, but they are important for fully understanding the underlying theory. Since, in reality, signals can be said to have a true beginning and end, we must admit that all signals we really work with in practice have infinite-bandwidth. That is, when a signal is turned on or off, there is a spectral event extending all the way to infinite frequency (while ``rolling off'' with frequency and having a finite total energy).E.2
In summary, audio signals are perceptually equivalent to bandlimited signals, and bandlimited signals are infinitely smooth in the sense that derivatives of all orders exist at all points time .
Points of Infinite Flatness