however, since a probability distribution is a bounded monotone function, ε is in some set S ∗ 1 $_{,d$_{c}$}$( V ) and thus theorem 4 requires only that all ε$_{n}$ be- long to S ∗ 1 $_{,d$_{c}$}$( V ), that (13) is satisfied for φ − 1 and that φ be differentiable in components of x. The condition (13) for φ − 1 is necessary for well-posedness: the example below shows that even in the weak topology of generalized functions well- posedness does not hold if it is not satisfied e.g. for deconvolution with supersmooth distributions. Example. Consider the function φ ( x ) = e − x $^{2}$, x ∈ R, then φ − 1 does not satisfy (13). Then there exists a sequence of functions γ$_{n}$ and a γ in S ∗ such that γ$_{n}$φ → γφ in S $^{∗}$, but γ$_{n}$ does not converge to γ in S $^{∗}$. Define the function b$_{n}$ ( x ) = φ − $^{1}$( x ) I ( n − 1 $_{n}$, n + 1 n ) ; then b$_{n}$ ∈ S $^{∗}$. Then define γ$_{n}$ = γ + b$_{n}$ for some fixed γ ∈ S $^{∗}$. Define ε$_{n}$ = γ$_{n}$φ and ε = γφ ; since φ ∈ S, the products are always defined. Consider the difference ε$_{n}$ − ε ; it equals I ( n − 1 $_{n}$, n + 1 n ) . In S ∗ this sequence converges to zero. On the other hand, ( γ$_{n}$ − γ, ψ ) for any ψ ∈ S equals Select ψ ∈ S given by ψ ( x ) = exp( − | x | ) ; then ( γ$_{n}$ − γ, ψ ) = ∫ n +2 /n n − 2 /n e x $^{2}$ψ ( x ) dx. ∫ n +2 /n n − 2 /n e x $^{2}$ψ ( x ) dx ≥ ∫ n +1 /n n − 1 /n e x $^{2}$− $^{x}$dx ≥ 1 2 n e − ( n + 1 $_{n}$)+ ( n − 1 $_{n}$) 2 . 40