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


This theorem requires a proof.
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.



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
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.