I would like to show that the Dirac distribution is not regular considering the test functions, which are infinitely differentiable and with compact support.
Proof 1
EDIT : what about this proof :
Assume on the contrary that there exists a locally integrable function $f$ on $\mathbb{R}$ such that for all test function $\varphi$,
$$
\varphi(0) = \int_{\mathbb{R}} f(x)\varphi(x) \, dx.
$$
Consider a compact $K\subset \mathbb{R}$ and $\psi_e = \chi_K * \phi_e$ where $\phi_e$ is a mollifier and $\chi_K$ is the characteristic function of $K$.
Then, $\psi_e \to \chi_K$ as $e\to 0$ and $\psi_e$ is a test function, hence
$$
\psi_e(0) = \int_{\mathbb{R}}f(x)\psi_e(x)dx
$$
and taking the limit as $e\to 0$ we obtain
$$
\chi_K(0) = \int_K f(x)dx.
$$
This being valid for any compact $K$, it is a contradiction.
Attempt 2 I also had the idea to use test functions $\varphi_e$ converging to the dirac distribution $\delta_0$ and such that $\int \varphi_e=1$. So we get something like $$ \varphi_e(0) = \int_{-c_e}^{c_e} f(x)\varphi_e(x)dx. $$ I can prove that $\varphi_e(0)\to \infty$ as $e\to 0$ but I can't prove that $\int_{-c_e}^{c_e} f(x)\varphi_e(x)dx$ is bounded (I expect that since I impose $\int \varphi_e=1$ and $f$ is locally integrable).
Is the first proof correct ?