 That is, formula_123.The cumulative distribution function of formula_64 is then

If function formula_126 is invertible (i.e., formula_127 exists, where formula_128 is formula_126's inverse function) and is either increasing or decreasing, then the previous relation can be extended to obtain

With the same hypotheses of invertibility of formula_126, assuming also differentiability, the relation between the probability density functions can be found by differentiating both sides of the above expression with respect to formula_132, in order to obtain

If there is no invertibility of formula_126 but each formula_132 admits at most a countable number of roots (i.e., a finite, or countably infinite, number of formula_136 such that formula_137) then the previous relation between the probability density functions can be generalized with

where formula_139, according to the inverse function theorem.The formulas for densities do not demand formula_126 to be increasing.