The (multivariate) Change of Variables Theorem (e.g. Blitzstein & Hwang 2019, Thm 8.1.7) states (loosely) that for a random vector X with continuous pdf $f_X$ and an invertible function $g:A_0\to B_0$, where $A_0$ and $B_0$ are open subsets of $\mathbb{R}^n$, the pdf of $Y=g(X)$ satisfies \begin{equation*} f_Y(y)\cdot |det(J)|=f_X( g^{-1}(y)) \end{equation*} where $J$ is the Jacobian of $g$.
Why is this Theorem restricted to functions whose domain $A_0$ is an open set? Can it be extended to include the boundary?
I am facing a random vector $X$ whose support are the non-negative reals and a function $g$ such that $g_i(x)=0$ if and only if $x_i=0$, and would like to have a statement relating $f_Y$ to $f_X$ for all non-negative real $X$.