Consider the equation $x F = 0$, where $F$ can be a generalized function. One family of solutions is of course delta functions $\alpha \delta$, where $\alpha \in \mathbb R$, and $\delta$ is the Dirac delta distribution. I came across the claim that these are the only solutions in Walter Appel's book Mathematics for Physicists. How do I prove it?
What we need to show, as far as I understand, is if $ (xF, \varphi)= (F, x \varphi) = 0$, where $\varphi$ is a test function (smooth and vanishes outside some compact interval), then $(F, \psi) = \alpha \psi(0)$ with some $\alpha \in \mathbb R$, for any test function $\psi$.
The first thing you can check is that any smooth compactly supported function $\varphi \in C^\infty_c$ whose support does not contain $0$ can be written as $x\psi$ for some $\psi \in C_c^\infty$, with $\psi = x^{-1} \varphi$.
It follows then that if $xF=0$, then the support (https://en.wikipedia.org/wiki/Support_(mathematics)#Support_of_a_distribution) of $F$ is precisely $\{0\}$. However, any point supported distribution is necessarily a linear combination of the Dirac delta at that point and its derivatives (https://ncatlab.org/nlab/show/point-supported+distribution). It is now a simple matter to check that derivatives of the Dirac can never satisfy your equation, i.e. for any linear combination which includes derivatives one can always construct a test function such that $\langle xF,\varphi\rangle\neq 0$. Thus, $F$ can only be $\alpha \delta_0$.