A distribution with zero derivative is constant

If $v^{\prime }=0$ on $\mathbb{R}$, then $v$ is a constant distribution. Every test function of integral zero is the derivative of its compactly supported primitive, so $v$ vanishes on those test functions. Choosing one test function $\eta$ of integral one gives $⟨v,\phi ⟩=⟨v,\eta ⟩\int \phi$. Iteration shows that a distribution with $m$th derivative zero is a polynomial of degree at most $m-1$.