Delta function at the boundary of a domain 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.

$\def\vr{{\bf r}}$
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 $n=10,100,1000$. 
