TITLE: Dirac Delta Function of a Function
QUESTION [45 upvotes]: I'm trying to show that
$$\delta\big(f(x)\big) = \sum_{i}\frac{\delta(x-a_{i})}{\left|{\frac{df}{dx}(a_{i})}\right|}$$
Where $a_{i}$ are the roots of the function $f(x)$. I've tried to proceed by using a dummy function $g(x)$ and carrying out:
$$\int_{-\infty}^{\infty}dx\,\delta\big(f(x)\big)g(x)$$
Then making the coordinate substitution $u$ = $f(x)$ and integrating over $u$. This seems to be on the right track, but I'm unsure where the absolute value comes in in the denominator, and also why it becomes a sum.
$$\int_{-\infty}^{\infty}\frac{du}{\frac{df}{dx}}\delta(u)g\big(f^{-1}(u)\big)
= \frac{g\big(f^{-1}(0)\big)}{\frac{df}{dx}\big(f^{-1}(0)\big)}$$
Can any one shed some light? Wikipedia just states the formula and doesn't actually show where it comes from.

REPLY [2 votes]: Normally, action of δ distribution on test function: (δ, φ) = ∫δ(x)φ(x)dx 
Now define the action of δ(g(x)): (δ(g(x)), φ) = lim ε→0(δ_ε(g(x)), φ) = lim ε→0 ∫δ_ε(g(x))φ(x)dx. This is just a definition to help us define what its action will be. 
This naturally converges to the action of δ distribution. As you probably remember from the action of the normal δ(x), it gives us φ(0) when integrated. So for δ_ε(g(x)) to have the appropriate property, its argument needs to be zero, which is where the roots of g(x) = 0 come in. 
In this case, the support of the test function φ needs to be in the neighborhood of  these roots: supp[φ(x)] = B_ε(a), where g(x) = 0 when x=a.
So now we're almost ready to completely define the action of δ_ε(g(x)):
(δ(g(x)), φ) = ∫δ_ε(g(x))φ(x)dx, where the region of integration is B_ε(a). Now I want to perform change of variables: y = g(x) near x=a. I only care about near x=a since the integration is only within the neighborhood of x=a
But for this change of variable to make sense, g'(x) cannot be changing signs from positive to negative, or vice versa. Because if it does, x = g_inverse(y) would not be a one to one relationship. --> We need g'(x) ≠ 0. If this is the requirement when we are working in 1D, for multivariate case, the corresponding requirement is |jacobian|x=a ≠ 0. Remember the absolute value is in place since it represents the volume of integration after the change of variable, and volume cannot be negative. 
Now we're ready for change of variable:
(δ(g(x)), φ) = ∫δ_ε(g(x))φ(x)dx = (1/|jacobian|x=a)*∫δ(y)φ(a)dy where the region of integration is B_ε(a). Here you can see that this is the action you would expect if 
(δ(g(x)), φ) = lim ε→0(δ_ε(g(x)), φ) = (1/|jacobian|x=a)*(δ(x-a), φ)