Using the Lorentzian as the delta function
$$\delta(x) ~=~ \lim_{\epsilon\rightarrow 0} \frac{1}{\pi}\frac{\epsilon^2}{\epsilon^2+x^2}$$
Is there a way to rigorously prove the sifting property, namely
$$\int^{\infty}_{-\infty} f(x) \delta(x-t) dx = f(t)$$