Can I use this Dirac delta distribution identity to evaluate this integral?

Question

I know of the identity $\int dy \delta(x-y)\delta(y-x') = \delta(x-x')$ but what about if I have an integral like $$\int dy \delta(x-y)\delta(y-x')f(y)$$Can the above identity be used in any way? Can say something like this?$$\int dx\int dy\delta(x-y)\delta(y-x')f(y) = \int dx \delta(x-x')f(x)$$

Answer 1

Yes. You can think of integrals with delta functions as this: the integral will only not be zero at x=x‘. You can plug the function g(x)=1 into the identity you listet and the result is g(x‘)* delta(x-x‘)=delta(x-x‘). Applying this to the second integral gives the result f(x‘)*delta(x-x‘)