is a.e. differentiable, ∂ $^{d$_{x}$}$F$_{x,y}$ ( x, y ) and f$_{x}$ ( x ) = ∂ $^{d$_{x}$}$F$_{x}$ ( x ) exist a.e. and f$_{x}$ ( x ) > 0 . When the density f$_{x}$ is a continuous function the conditional distribution can be represented as a functional on a function space on R d$_{x}$ that can be derived from the general representation above in D $^{∗}$( W ) . Indeed, any distribution function, F ( x, y ) , where we focus on the ar- gument x, via the copula representation can be considered as a functional on D ( W ) . Let Φ denote the class of such distribution functions, then Φ ⊂ D $^{∗}$( W ) . Moreover the representation (12) demonstrated that any conditional distribution F$_{|}$$_{x}$ ( x, y ) also defines a linear continuous functional on D ( W ) . Denoting by Φ$_{|}$$_{x}$ the class of conditional distributions we thus have shown that Φ$_{|}$$_{x}$ ⊂ D $^{∗}$( W ) . By the remark, we can relax the differentiability condi- tions and consider Φ$_{|}$$_{x}$ ⊂ D ∗ k ( W ) ; when the distribution function is differ- entiable in x, we set k = 0 . On the other hand, then a continuous density function, f$_{x}$ > 0 exists and the conditional distribution can be represented by an ordinary function ∂ d x F$_{x,y}$ ( x,y ) f$_{x}$ ( x ) ; denote by Φ$_{c}$ the class of distributions that are continuously differentiable in x with f$_{x}$ > 0 on R $^{d$_{x}$}$, and by Φ$_{c}$$_{|}$$_{x}$ the class of corresponding conditional distributions. Then Φ$_{c}$ ⊂ D ∗ $_{0}$( R $^{d$_{x}$}$) and as well Φ$_{c}$$_{|}$$_{x}$ ⊂ D ∗ 0 ( R d$_{x}$ ) , where the space D$_{0}$ ( R d$_{x}$ ) is the space of continuous functions with bounded support in R $^{d$_{x}$}$. Since Φ$_{c}$$_{|}$$_{x}$ ⊂ Φ$_{|}$$_{x}$ , any conditional distribution that exists in the ordinary sense and thus is in Φ$_{c}$$_{|}$$_{x}$ , has two representations: one as a functional on D$_{0}$ ( W ) defined above and the second 20