The 
or impulse ``function'' is not, strictly speaking, a function at
all, but is a generalized function. A generalized function is defined in
terms of the class of all equivalent regular sequences of particularily
well-behaved functions. A particularily well-behaved function 
is
bounded by 
as for any large N as 
. A regular
sequence 
of well-behaved functions have the property that
exists for any well-behaved function 
. Strictly speaking, this definition
precludes simple sequences such as 
as 
goes to 0
(which is often used to develop the 
function) since 
is not well-behaved. However, so long as the derivative is not
required, this approach is useful. For further information see Bracewell
[].
It is important to note that a generalized function is not defined in
terms of the limit of a single sequence of functions but of a class of
equivalent functions. For example, the 
function may be
(loosely) defined in terms of the limit of 
as 
goes to 0
or equivalently in terms of the limit of 
as 
goes to 0.
Note that whenever a generalized function 
appears in an integral, the
integral should be interpreted as the limit of the integral, i.e.,
should be interpreted as
where 
is the sequence defining 
. Hence, the Fourier transform
is a limit for the case of a generalized function.
A particular property of the generalized function 
is the
so-called ``sifting'' or ``sampling'' property, i.e.,
Note that whenever a generalized function appears in an integral the integral should be intepreted as a limit; hence, this equation should be understood to be,
where 
is the defining class of the regular sequence of
well-behaved functions.
Using these relatively obscure concepts invented to permit mathematicans work
with the generalized functions (the 
in particular) so useful in
engineering, we can easily compute the derivative of the 
function (the
doublett, denoted 
) and its integral (the unit step function
). We can also determine that 
is even
(
) and that
Using the sifting property of the 
function is can be easily seen that
. Based on the symmetry property of the
Fourier Transform, it follows that 
(
definition) or 
(f definition).