Null Function

A null function $\delta^0(x)$ satisfies

$$\int^b_a \delta^0(x)\,dx = 0$$ (1)

for all $a, b$, so

$$\int_{-\infty}^\infty \vert\delta^0(x)\vert\,dx = 0.$$ (2)

Like a Delta Function, they satisfy

$$\delta_0(x) = \cases{ 0 & $x \not= 0$\cr 1 & $x = 0$.\cr}$$ (3)

See also Delta Function, Lerch's Theorem