This wikipedia article https://en.wikipedia.org/wiki/Probability_density_function has the formula ${\displaystyle f_{Y}(y)=\int _{\mathbb {R} ^{n}}f_{X}(\mathbf {x} )\delta {\big (}y-V(\mathbf {x} ){\big )}\,d\mathbf {x}}$ for density for random variable $Y=V(X)$, for a differentiable function $V$.
Since $\delta$, which is supposed to be the delta function, it should be a distribution acting on $f_X$.
What confuses me is it is a composition with a function. Since the integral should be only formal, and thus $\mathbf{x}$ is only formal, I am not sure how this distribution is defined. I guess distributional derivative of some sort of Heaviside function?
What kind of distribution is this? There are many questions on compositions of delta function on this website, but they seem to consider formal calculations.
I think my problem is I don't know how the functional $\delta(y-V(\cdot))$ is defined. The very definition seems to take the variable $\mathbf{x}$
They mean $$\int_{\Bbb{R}^n} f(x)\delta(y-V(x))dx= \lim_{n\to \infty} \int_{\Bbb{R}^n} f(x)2n\, 1_{|y-V(x)|< 1/n}dx$$
$\delta(t)$ is the distribution which is the limit in the sense of distributions of the sequence of functions $2n \, 1_{|t|<1/n}$.
Writing distributions as limits of sequences of distributions makes composition with functions, change of variables, differentiation (and so on) meaningful and consistent.