Could someone explain to me why $\displaystyle \int_{-\infty}^\infty f(x)\delta(x)\, dx = f(0) $.
Well, to be honest with you I am no expert on this field. But, I was under the impression that the integral you quoted is an abuse of notation considering the way $\displaystyle \delta$ is defined makes it impossible for $\displaystyle \int_{-\infty}^{\infty}f(x)\delta(x)dx$ to be an actual Riemann or Lebesgue integral.