Find all function $f\in C^{\infty}(\mathbb{R})$, such that $f\delta^{'}=0$ ,where $\delta^{'}$ is a distribution known as the dipole distribution defined as follows $$\delta^{'}:D(\mathbb{R})\to \mathbb{R}$$ as $\delta^{'}(\phi)=\phi^{'}(0)$.
My attempt: We have $f\delta^{'}(\phi)=0$ for all $\phi \in D(\mathbb{R})$. So by definition, we have $\delta^{'}(f\phi)=0 \implies (f\phi)^{'}(0)=0 \implies f^{'}(0)\phi(0)+ \phi^{'}(0)f(0)=0 $. Now we can choose $\phi \in D(\mathbb{R})$ such that $\phi(0)=0, \phi^{'}(0)=1$, so that gives $f(0)=0$. Now next what I have in mind is that I can choose another $\phi \in D(\mathbb{R})$ such that $\phi(0)=1, \phi^{'}(0)=0$. Then it will gives $f^{'}(0)=0$. So I have all those $f\in C^{\infty}(\mathbb{R})$ such that $f(0)=0, f^{'}(0)=0$. Can I have a nicer characterization of $f$, a more nice looking family? Also I want to know whether my first choice of $\phi$ is valid or not?
Thanks in advance!