Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 327 / 2 / a / Solution

If a test function $\phi$ has integral zero, thenis another compactly supported smooth test function with ${\psi }^{\prime }=\phi$. Hence $v^{\prime }=0$ implies $⟨v,\phi ⟩=-⟨v^{\prime },\psi ⟩=0$ for every such $\phi$.

Choose a test function $\eta$ with integral one. For arbitrary $\phi$, the function $\phi -\eta \int \phi$ has zero integral, soThus $v$ is the constant distribution $⟨v,\eta ⟩$. This proves a distribution with zero derivative is constant.