Delta function at the boundary of a domain

Question

What is the rigorous definition of a Dirac delta function that evaluates test functions at the boundary of their domain? For example, using spherical coordinates, how do I rigorously interpret the following equation $$\int_0^\infty f(r) \delta(r) d r = f(0)$$ for all test functions $f(r)$ defined for $r \in [0, \infty \rangle$? Equations of this sort are common in physics, but never really justified.

Answer 1

$\def\vr{{\bf r}}$ Consider the integral $$I = \int_{\mathbb{R}^3} g(\vr)\delta(\vr-\vr_0)dV.$$ A standard delta sequence on $\mathbb{R}$ is $$\delta_n(x)=\sqrt{\frac{n}{\pi}}e^{-nx^2}$$ so we write \begin{align*} I_n &= \int_{\mathbb{R}^3} g(r) \left( \prod_{i=1}^3 \sqrt{\frac{n}{\pi}}e^{-n(x_i-x_{i0})^2} \right) dV \\ &= \int_{\mathbb{R}^3} g(r) \left(\frac{n}{\pi}\right)^{3/2} e^{-n(\vr-\vr_0)^2} dV. \end{align*} (By assumption, $g(\vr)=g(r)$.) If $\vr_0={\bf 0}$, then \begin{align*} I_n &= \int_0^{2\pi}\int_0^\pi\int_0^\infty g(r) \left(\frac{n}{\pi}\right)^{3/2} e^{-nr^2} r^2\sin\theta dr d\theta d\phi \\ &= \int_0^\infty g(r) \frac{4n^{3/2}}{\sqrt{\pi}} r^2 e^{-nr^2} dr. \end{align*} Thus, $$\delta_n(r) = \frac{4n^{3/2}}{\sqrt{\pi}} r^2 e^{-nr^2}$$ must be a delta sequence for $\delta(r)$.

Figure 1. $\delta_n(r)$ for $n=10,100,1000$.