Dirac Delta Function

The Dirac Delta Function at $t_{0}, δ (t - t_{0})$ , is defined by the property

\int_{0}^{\infty} f (t) δ (t - t_{0}) d t = f (t_{0})

for any continuous function $f (t)$ .

$δ$ is not a function, but more of a general object called a Distribution

Thus,

\begin{aligned} L {f (t) δ (t - t_{0})} & = \int_{0}^{\infty} e^{- s t} f (t) δ (t - t_{0}) d t \\ = e^{- s t_{0}} f (t_{0}) \end{aligned}

Corollary: