I once learned that when computing the density $f$ of $g(x)$ (where X is some continuous random variable) the transformation $g$ needs to be continuous, injective and differentiable in order to use the usual Jacobian transformation.
Is it enough if the transformation is almost everywhere differentiable? I saw it used in a paper where $g$ was defined piecewise (it was continuous but not continuously differentiable), implying that the density $f(g)$ would not be defined at every point that $g(x)$ could attain.
How would one derive the probability density function at that point? At least in some cases, the pdf should exist (e.g. folded normal distribution at $x=0$). How do I find out if it does exist and how do I compute it if the Jacobian transformation is not applicable (which I think it is not).