What Are the Key Properties of the Dirac Delta Function?

[arierreF]

Prove that.

$\int_a^b f(x)g' (x)\, dx = -f(0)$


This is supposed to be a delta Dirac function property. But i can not prove it.
I thought using integration by parts.

$\int_a^b f(x)g' (x)\, dx = f(x)g(x) - \int_a^b f(x)'g (x)\, dx$

But what now?


Some properties:


$\delta [g(x)] = \sum \frac{1}{|g'(xi)|}$

$\int_a^b f(x)\delta(x-xi)\, dx =$

$f(x_{0})$ if $a<x_{0}<b$
0, other cases.




I just need a tip please.

 

[mfb]

Prove that.

$\int_a^b f(x)g' (x)\, dx = -f(0)$

In general, this is wrong. Are there any additional constraints on f,g,a,b?
If that would be true, all integrals would be trivial ;).

 

[HallsofIvy]

What does this have to do with the "Dirac Delta Function"? Is g' supposed to be the Dirac Delta Function? What is g?