$\textbf{Exercise.}$ Let be $u \in \mathfrak{D}'(\Omega)$ and $\varphi \in \mathfrak{D}$ such that $\varphi|_{\text{supp} \ u} \equiv 0$. Is it true that $\varphi u \equiv 0$?
It seems obviously true that $\varphi u \equiv 0$, but I would like to check my attempt because this exercise seems extremely simple to be an exercise. If I'm wrong, I would like a hint in order to construct a counterexample because I can't see how $\varphi u \neq 0$ in these hypothesis.
What I thought is pretty simple:
Case $1$ - $x \in \text{supp} \ u$:
$(\varphi u) (x) = \varphi(x) u(x) = 0 u(x) = 0$.
Case $2$ - $x \notin \text{supp} \ u$:
$(\varphi u) (x) = \varphi(x) u(x) = \varphi(x) 0 = 0$,
therefore $\varphi u \equiv 0$. $\square$
Take $\phi\in\mathcal{D}$. Then by definition of multiplication of smooth function and distribution:
$$\varphi u(\phi) \equiv u(\varphi \phi)$$
Can you take it from here, using the definition of $\text{Supp}(u)$?