Lemma: A bounded bulk term cannot cancel a surface delta

Let $c$ be locally bounded near $\Sigma$, and let $\beta$ be smooth on $\Sigma$. If $\beta ({\mathbf{z}}_0)\ne 0$, then

is not zero in any neighbourhood of $(0,{\mathbf{z}}_0)$.

Proof. Choose a spatial test $\psi$ supported arbitrarily near ${\mathbf{z}}_0$ with $\int \beta \psi \ne 0$. Pair with $\chi (t/ϵ)\psi (\mathbf{z})$, where $\chi (0)=1$ and $\chi$ is compactly supported. The delta pairing is fixed and nonzero. Local boundedness makes the bulk pairing $O(ϵ)$. For sufficiently small $ϵ$ they cannot cancel.