Jump Rule

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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:

\(\ds \map {T_f'} \phi\) \(=\) \(\ds - \map {T_f} {\phi'}\) Definition of Distributional Derivative
\(\ds \) \(=\) \(\ds - \int_a^b \map f x \map {\phi'} x \rd x\)
\(\ds \) \(=\) \(\ds - \int_a^c \map f x \map {\phi'} x \rd x - \int_c^b \map f x \map {\phi'} x \rd x\)
\(\ds \) \(=\) \(\ds \map f a \map \phi a - \map f {c^-} \map \phi {c^-} + \int_a^c \map {f'} x \map \phi x + \map f {c^+} \map \phi {c^+} - \map f b \map \phi b + \int_c^b \map {f'} x \map \phi x \rd x\) Integration by parts
\(\ds \) \(=\) \(\ds \int_a^b \map {f'} x \map \phi x \rd x + \paren {\map f {c^+} - \map f {c^-} } \map \phi c\) Definition of Test Function
\(\ds \) \(=\) \(\ds \map {T_{f'} } \phi + \paren {\map f {c^+} - \map f {c^-} } \map {\delta_c} \phi\) Definition of Dirac Delta Distribution
\(\ds \) \(=\) \(\ds \map {\paren {T_{f'} + \paren {\map f {c^+} - \map f {c^-} }\delta_c} } \phi\) Distribution is linear mapping

$\blacksquare$


Also see


Sources