As mentioned in §3.6, every audio signal can be regarded as
infinitely differentiable due to the finite bandwidth of human
hearing. That is, given any audio signal , its Fourier
transform is given by

For the Fourier transform to exist, it is sufficient that be
absolutely integrable, i.e.,
. Clearly, all audio
signals in practice are absolutely integrable. The inverse Fourier
transform is then given by

Because hearing is bandlimited to, say, kHz, sounds
identical to the bandlimited signal

where
. Now, taking time derivatives is simple (see also §C.1):

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
.