Jump Rule

Theorem
Let $f : \R \to \R$ be a piecewise continuously differentiable real function with a discontinuity at $c \in \R$.

Suppose the limits $\map f {c^+}, \map f {c^-}, \map {f'} {c^+}, \map {f'} {c^-}$ exist.

Let $T \in \map {\DD'} \R$ be a distribution associated with $f$.

Then in the distributional sense we have that:


 * $T_f' = T_{f'} + \paren {\map f {c^+} - \map f {c^-}} \delta_c$

where $\delta_c$ is the Dirac delta distribution.

Proof
Let $\phi \in \map \DD {\R}$ be a test function with its support on $\closedint a b \subset \R$.

Let $c \in \closedint a b$.

Then:

Also see

 * Definition:Jump Discontinuity