 Solution

The distributional derivative is defined by $⟨F^{\prime },\varphi ⟩=-⟨F,{\varphi }^{\prime }⟩$ for every test function. For the Heaviside function, compact support givesThus ${\mathcal{H}}^{\prime }=\delta$. Splitting the integral for $|x|$ at zero and using integration by parts gives its first derivative as the sign function. The jump of that function at zero is two, soas distributions. The value assigned to the sign function at the single point zero does not affect its associated distribution.