Distributional Derivative of Absolute Value Function

Theorem
Let $H: \R \to \closedint 0 1$ be the Heaviside step function.

Let $\size x$ be the absolute value of $x$.

Let $T_{\size x}$ be the distribution associated with $\size x$.

Then the distributional derivative of $T_{\size x}$ is $T_{2 H - 1}$

Proof
Furthermore:


 * $\ds \lim_{x \mathop \to 0^+} \size x = \lim_{x \mathop \to 0^-} \size x = 0$

By the Jump Rule:


 * $T_{\size x}' = T_{2H - 1}$

Also see

 * Derivative of Absolute Value Function