The continuous-time impulse response was derived above as the
inverse-Laplace transform of the transfer function. In this section,
we look at how the impulse itself must be defined in the
continuous-time case.

An impulse in continuous time may be loosely defined as any
``generalized function'' having ``zero width'' and unit
area under it. A simple valid definition is

(C.5)

More generally, an impulse can be defined as the limit of
any pulse shape
which maintains unit area and approaches zero width at time 0. As a
result, the impulse under every definition has the so-called
sifting property under integration,

(C.6)

provided is continuous at . This is often taken as the
defining property of an impulse, allowing it to be defined in terms
of non-vanishing function limits such as

An impulse is not a function in the usual sense, so it is called
instead a distribution or generalized function
[13,44]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)