An impulse in continuous time must have ``zero width''
and unit area under it. One definition is

(B.3)

An impulse can be similarly defined as the limit of any
integrable 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,

(B.4)

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

(Note, incidentally, that
is in but not
.)

An impulse is not a function in the usual sense, so it is called
instead a distribution or generalized function
[10,37]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)