as is the case when F is a probability distribution function, then it can be computed by integration: For the function F (5) the functional on the right-hand side defines the generalized derivative: f = ∂ $^{k}$F ∂x 1 $_{...∂x$_{k}$}$. First consider density as a generalized function on the space D$_{∞}$ ( W ) . Theorem 1. The inverse problem (1) for any cumulative probability distribution function F has the solution f defined by (5) in the space of gen- eralized functions D ∗ for D$_{∞}$ ( W ) . The problem is well-posed. When density exists as an integrable function, f ( x ) , it provides the generalized function f via the value of the corresponding functional: Proof. Any distribution function F on R k is a monotone bounded function and as such is locally integrable on any bounded set; a function like that represents a regular element in the space of generalized functions, D $^{∗}$, for D$_{∞}$ ( W ) defined above. Then (5) defines f as the generalized derivative of F, the generalized density function. The differentiation operator ∂ k = ∂ ∂x 1 $_{...∂x$_{k}$}$on the space of generalized func- tions D ∗ is defined for any regular function and is a continuous operator 11 ( f, ψ ) = ∫ ... ∫ f ( x$_{1}$, ..., x$_{k}$ ) ψ ( x$_{1}$, ..., x$_{k}$ ) dx$_{1}$...dx$_{k}$. (6) ( F, ∂ $^{k}$ψ ∂x$_{1}$...∂x$_{k}$ ) = ∫ ... ∫ F ( x$_{1}$, ..., x$_{k}$ ) ∂ $^{k}$ψ ( x$_{1}$, ...x$_{k}$ ) ∂x$_{1}$...∂x$_{k}$ dx$_{1}$...dx$_{k}$.