Distributional Derivatives of Dirac Delta Distribution do not Vanish

Theorem
Let $\delta \in \map {\DD'} \R$ be the Dirac delta distribution.

Then for any $n \in \N$ the distributional derivative $\delta^{\paren n}$ does not vanish.

Proof
Let $\phi \in \map \DD \R$ be a test function such that $\map \phi 0 \ne 0$.

Then:


 * $\forall n \in \N : \forall x \in \R : x^n \phi \in \map \DD \R$

By the definition of the distributional derivative: