Proposition: Fixed-support scalar test continuity

For fixed compact supports of $f,g,f^{\prime }$, the map $(f,g,f^{\prime })\mapsto {\mathcal{N}}_c(f,g,f^{\prime })$ is jointly continuous. The same holds for each fixed observation descendant.

Proof. A distribution has a finite-order estimate on every fixed compact support. Apply that estimate to $f\otimes g\otimes f^{\prime }$. Leibniz's rule bounds its test seminorms by products of finitely many smooth seminorms of the three factors. Fixed differential operators add only finitely many required test derivatives.