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The Cauchy approximate identity isIt is a probability density of total mass one, and its mass outside every fixed neighbourhood of zero tends to zero as $\epsilon \to 0$. It therefore converges to the Dirac delta function as a distribution.

For $L=D_t^2+2k{D}_t+k^2+{\omega }^2$, the causal Green function satisfies $LG=\delta (t)$. It is continuous at zero, with a unit jump in its first derivative. Its smooth positive-time kernel $g$ obeys $g(0)=0,g^{\prime }(0)=1$. The zero-frequency limit is $H(t)t{e}^{-kt}$. Shifting $G$ gives the zero-initial-data response to a delayed Dirac delta function.

Gaussian white noise is a generalized Gaussian process with zero mean and covariance proportional to a Dirac delta function in time, and possibly in space. It is the formal time derivative of Brownian motion and has independent fluctuations on disjoint time intervals.

A sequence can converge as distributions off one point while retaining a nonconvergent multiple of a Dirac delta function at that point. If $u_j=v+c_j{\delta }_p+o_{{\mathcal{D}}^{\prime }}(1)$ with $c_j$ nonconvergent, restriction to the punctured domain tends to $v$, but a test nonzero at $p$ detects the obstruction to global convergence.

A narrow surface density of a disk source approaching the outer edge from inside, with $S=S_0\delta (r-r_2)$ normalized to have full weight, gives $\dot{M}=2\pi r_2{S}_0$ and the displayed steady interior profile. This follows by replacing the inner source integral by $r_2{S}_0$ and integrating $u^{-1/2}$. The flux jumps from $\dot{M}$ just inside to zero outside the injecting edge. Thus a zero exterior flux is consistent, whereas a continuous zero interior boundary trace is not. A half-weight convention for a symmetric endpoint Dirac delta function represents a different total injected mass; it must not be silently mixed with the full-weight boundary source.