### The Continuous-Time Impulse

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

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,

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*
[12,38]. (It is still commonly called a ``delta function'',
however, despite the misnomer.)

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