Sorry for I'm such a beginner of mathematics that there maybe something trival is missed. This looks terrible as a mixture of the language of analysis and the one of algebra. Whereas it appeals to me for the physical picture lying behind it.
My main problem is that, if we consider (all those distributions over $V$) $\mathcal D'(V)$ as a module over (all those compact support smooth / continuous functions over $V$) $\mathcal C(V)$ , we will have an exact sequence: $0 \rightarrow \mathcal C(V) \xrightarrow{i} \mathcal D' (V)$, where the inclusion defined as $i(f) := \int_V f\cdot(-) \mathrm d \mu$.
Therefore I'm wondering about whether the sequence has a retraction, or similarly, the exact sequence $\mathcal D' (V) \xrightarrow{p} \mathcal D' (V) / \mathcal C(V) \rightarrow 0$ is split or not.
So does the retraction $r$ which is continuous and a homomorphism of module exist? It is also acceptable if $r$ is simply a homomorphism of module.
This problem arised from the mathematical description of point charge $\delta$, we may consider it as a small ball, in order to ensure the electric potential is continuous. (but not the limit of testing function!) In this case, the retraction is the mapping sending $\delta$ into $1_{B(0, r)}$, the characteristic function over $B(0, r)$.
I dreamed that $r = (\alpha \mid 1_{B(x, r)})$, where $(\mid)$ is the pairing, is the solution. However sadly and evidently, this is not injective over $\mathcal C (V)$. Is there any possibility to adjust it into a retraction?