I had this on an exam yesterday, and I'm not entirely convinced that the statement is true. We were asked to show that the function $\delta (x) = \int_{-∞}^{∞} \frac{1}{t(t-x)} dt$ is a dirac delta function by demonstrating that $I=\int_{-∞}^{∞} f(x)\delta(x) dx$ holds all the necessary properties.
There are three things I believe should be shown: 1) The function should be infinite at a single point. (this function is infinite at $t=x$) 2) It should be zero everywhere else 3) It should satisfy $\int_{-∞}^{∞} f(x)\delta(x) dx=f(0)$.
I showed 2 is true by demonstrating that the Cauchy Principal Value is zero for that integral which means that it's zero everywhere save that one point we avoid.... but I don't see how 3 holds in general. I see that it holds for some functions, like $f(x)=x$, but what about $f(x)=1$, for example. Anyway, is this a delta function or not.... if so, why?