Jump Rule/Finite Number of Discontinuities

Theorem

Let $f : \R \to \R$ be a piecewise continuously differentiable real function with discontinuities at $\set {c_0, \ldots, c_n}$ where $c_k \in \R$.

Suppose for all $k \in \N_{\mathop \le n}$ the limits $\map f {c_k^+}, \map f {c_k^-}, \map {f'} {c_k^+}, \map {f'} {c_k^-}$ exist.

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

Then in the distributional sense we have that:

$\ds T_f' = T_{f'} + \sum_{k \mathop = 0}^n \paren {\map f {c_k^+} - \map f {c_k^-}} \delta_{c_k}$

where $\delta_c$ is the Dirac delta distribution.

Proof

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Also see

• Definition:Jump Discontinuity

Sources

• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 6.2$: A glimpse of distribution theory. Derivatives in the distributional sense