(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I would like to prove that [tex]{d \over {dx}} \sigma(x)=2\delta(x)[/tex],

where [tex]\sigma(x>0)=1[/tex]

[tex]\sigma(x=0)=0[/tex]

[tex]\sigma(x<0)=-1[/tex]

2. Relevant equations

3. The attempt at a solution

I think that I need to show that

[tex]\int_{-\infty}^{\infty}{d \over {dx}} \sigma(x)dx=2=\int_{-\infty}^{\infty}2 \delta(x)dx[/tex].

The integral looks rather harmless, and I would like to write

[tex]\int_{-\infty}^{\infty}{d \over {dx}} \sigma(x)dx=\sigma(x) |^{\infty}_{-\infty}=1-(-1)=2[/tex].

This looks like it works, but it seems to me that there should be some complication when integrating close to zero where we have discontiuous behaviour.

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# Integrating over a discontinuity

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