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## Homework Statement

[itex] g(x) [/itex] is a function with a discontinuity at [itex] x_0 [/itex] s.t.,

[tex] \Delta g_0 = \lim_{ \epsilon \to 0} (g(x_0 + \epsilon) - g(x_0 - \epsilon) ) [/tex]

## The Attempt at a Solution

I'd like to show that the following limit,

[tex] \lim_{\epsilon \to 0} \int_{x_0-\epsilon}^{x_0+\epsilon}g'(x)\varphi(x)dx = \Delta g_0 \varphi(x_0) [/tex]

where [itex] \varphi(x) [/itex] is some smooth test function that vanishes at [itex] \infty [/itex].

Intuitively I know this makes sense, but I'm having trouble showing it formally - any ideas/tips/advice?

edit: corrected mistake in original post.

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