Note that although support of the random y belongs to R d$_{y}$ it could be a discrete set of points, thus we do not restrict y to be continuously distributed. Consider the copula function (Sklar, 1973): C$_{F}$$_{x}$$_{,F}$$_{y}$ ( a, b ) on W = (0 , 1) 2 that is identical to the joint distribution function, that is for the mapping M : R $^{d$_{x}$}$× R d$_{y}$ → W defined by { x, y } → { F$_{x}$ ( x ) , F$_{y}$ ( y ) } we get the corresponding mapping M $^{∗}$( F$_{x,y}$ ( x, y )) = C$_{M}$$_{(}$$_{x,y}$$_{)}$ ( M ( x, y )) with Thus (10) is equivalent to denote F$_{x}$ ( x + Δ) − F$_{x}$ ( x ) by ˜ Δ , then by Assumption 2, continuity of F$_{x}$, Δ → 0 implies ˜ Δ → 0 thus the limit is equivalent to Since with respect to its second argument the copula function and the limit are ordinary functions we concentrate on being able to define the gen- eralized derivative with respect to the first argument. In particular, for any ψ ∈ D ( W ) , given the second argument the value of the functional (( C$_{F}$$_{x}$$_{,F}$$_{y}$ ) ′ $_{1}$, ψ ) = − ( C$_{F}$$_{x}$$_{,F}$$_{y}$, ψ ′ ) . This implies that we can define the value 17 lim ˜ Δ → 0 C$_{F}$$_{x}$$_{,F}$$_{y}$ ( a + ˜ Δ , b ) − C$_{F}$$_{x}$$_{,F}$$_{y}$ ( a, b ) ˜ Δ . F$_{y}$$_{|}$$_{x}$ = lim Δ → 0 C$_{F}$$_{x}$$_{,F}$$_{y}$ ( F$_{x}$ ( x + Δ) , F$_{y}$ ( y )) − C$_{F}$$_{x}$$_{,F}$$_{y}$ ( F$_{x}$ ( x ) , F$_{y}$ ( y )) F$_{x}$ ( x + Δ) F$_{x,y}$ ( x ) ; C$_{M}$$_{(}$$_{x,y}$$_{)}$ ( M ( x, y )) = C$_{F}$$_{x}$$_{,F}$$_{y}$ ( F$_{x}$ ( x ) , F$_{y}$ ( y )) = F$_{x,y}$ ( x, y ) .