Let $\varphi : \mathbb{R} \rightarrow \mathbb{R}$ be an infinitely differentiable function of compact support. Define $$\psi (x) := \begin{cases} \frac{\varphi (x) - \varphi (0)}{x}, & \text{$x \neq 0$} \\ \varphi'(0), & \text{$x = 0$} \end{cases}. $$
Is $\psi$ be infinitely differentiable and compactly supported?
Some context: $\psi$ is given as a hint to prove that for a distribution $f$ with the property $xf = 0$ (in the sense of distributions), $f = C \delta_0$, for some constant $C$. If $\psi$ has the appropriate properties, then the proof for the main question follows quickly. After some tinkering, however, I am not convinced that $\psi$ has either necessary property.