One of the properties of the dirac delta function is:
$$\int_{-\infty}^{\infty}\delta(x-a)f(x)dx=f(a)$$ Where here I am using the notation $\delta(x-a)$ to mean the spike at $x=a$
With all rigor, "it's not actually a function", and other arguments like this aside, I was wondering if the following approximation is ever valid:
$$\int_{x_0}^{\infty}\delta(x-a)f(x) \approx f(a)$$
Especially when using $$\delta(x-a)\approx \frac{e^{-(\frac{x-a}{b})^2}}{|b|\sqrt\pi}$$ When $b$ is made sufficiently small and $x_0<a$? Otherwise, the integral would be $0$ if $x_0>a$?
I have done a few numerical calculations with a variety of functions and it seems to hold up pretty well. Presumably if $f$ is particularly misbehaved then maybe this could have some problems, but I am not looking for functions that only exist as some counterexample in an analysis text. I am wondering whether or not this approximation is valid for well-behaved functions.