Composition of Dirac Delta Distribution with Function with Simple Zero/Proof 2

Proof
Let $H$ be the Heaviside step function.

Let $T \in \map \DD \R$ be a distribution associated with $\map H {\map f x}$:


 * $\ds T = T_{\map H {\map f x}}$

We have that:

Taking the derivative of the yields:

Taking the derivative of the yields:


 * $\ds \forall x \ne x_0 : \dfrac {\d} {\d x} \map H {\map f x} = 0$

Furthermore:


 * $\forall x \in \R : \map {f'} x > 0 : \paren {\map f {x_0^+} = 1} \land \paren {\map f {x_0^-} = 0}$


 * $\forall x \in \R : \map {f'} x < 0 : \paren {\map f {x_0^+} = 0} \land \paren {\map f {x_0^-} = 1}$

By Jump Rule the reads:

Define the composite Dirac delta distribution according to Distributional Derivative of Heaviside Step Function.

We have that:


 * $\ds T_{\map H x}' = \delta_0 = \delta_{x}$

Then:

Hence:

and: