Take the following integral: $$\int^{\infty}_{-\infty}f(x)\delta(cx)dx$$ Why can I not say that the argument of $\delta(cx)$ "picks out" the value $x=0$, making the integral $f(0)$? By "picks out", I am trying to use the rule that we only care about the value in the range of the bounds that makes the argument zero.
I know that I can do a change of variable, so I actually get that the integral evaluates to $\frac{1}{c}f(0)$, but I am not sure what is wrong with the initial claim.