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?