A distribution with zero derivative is constant
= A distribution with zero derivative is constant
If $v'=0$ 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 $\langle v,\varphi\rangle=\langle v,\eta\rangle\int\varphi$. Iteration shows that a distribution with $m$th derivative zero is a <polynomial> of degree at most $m-1$.