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Dirac delta function

An idealized instantaneous unit-area kick, concentrated entirely at one instant.

The Dirac delta $\delta(t-t_0)$ is the limit of a unit-area pulse narrowing to zero width at $t_0$ ; for any continuous $f$ , $\int_0^\infty f(t)\delta(t-t_0)\,dt=f(t_0)$ .