I just start reading the book "distributions theory and applications" in order to understand the notion of generalized function and densities
The following statement is from that book:
"In undergraduate physics a lecturer will be tempted to say on certain occasions: Let $\delta(x)$ be a function define on the line that equals 0 away from 0 and is infinite at 0 in such a way that it's total integral is 1. the most important property of $\delta(x)$ is exemplified by the identity
$\int\limits_{- \infty}^{+ \infty} \phi(x)\delta(x)dx= \phi(0) \qquad ... (*)$,
Where $\phi$ is a continuous function of x ".
Why is the equation (*) true ?
Could you please explain or point me to a reference wich explain the notion of "test density" in a context other then the probability theory, I've search for it but all find is the notion of test function and the notion density in probability!
Thanks!