delta function

i.e. Dirac delta function, unit impulse

$\delta(x)=\infty$ , if $x=0$ , else (when $x \neq 0$ ), $\delta(x)=0$ . $\int_{-\infty}^{\infty} \delta(x) \, dx = 1$

a fundamental property of the delta function:

\int_{-\infty}^\infty f(x) \delta(x-a)\, dx = f(a)

and for $\epsilon > 0$ ,

\int_{a-\epsilon}^{a+\epsilon} f(x) \delta(x-a)\, dx = f(a)

\begin{aligned} \delta(x-a)=0 \\ \delta(a \, x) = \frac{1}{|a|} \delta(x) \\ \delta(x^2 - a^2) = \frac{1}{2|a|}[\delta(x+a) + \delta(x-a)] \end{aligned}

\mu_0(t) \triangleq \delta(t)

with impulse/Dirac delta

\int_{-\infty}^t f(\lambda) \delta(\lambda - \tau)d\lambda = \begin{cases}0 \quad \text{for } t < \tau \\ f(\tau) \quad \text{for } t > \tau \end{cases}

Sifting property:
Sifting property#Continuous form

References: