@zed111 The formal underpinning of stuff like $\delta$ "functions" is the theory of
distributions. Once you have established what a distribution is and what it means to take the Fourier transform of it, the derivation is as easy as saying that $\int f(t)\delta(t) \mathrm{d}t = f(0)$ is the defining property of $\delta(t)$ and showing that the integral you're describing has precisely that property as a distribution in $t$