This assumption does not exclude the possibility that g has some singular- ities, for example, it could be a sum of δ − functions or a mix of a regular function and some peaks represented by δ -functions. Denote by f$_{z}$ the density of z. Recall that f denoted the generalized density of u and γ = F t ( g ) ; φ = F t ( f ) , ε$_{·}$ = F t ( w$_{·}$ ) . (a) supp ( γ ) is a connected set that includes 0 as an interior point; (b) supp ( φ ) = R $^{d}$. (c) supp ( f$_{z}$ ) = R $^{d}$. Assumption 5(a) would not be satisfied by a polynomial regression func- tion when support of γ consists of one point 0; semiparametric estimation for that case was provided in Hausman et al (1991). Support assumptions (b) and (c) are standard; they could be relaxed. Assumption 5 (support). Recall that here φ is a characteristic function; φ (0) = 1 . (a) The function f$_{z}$ is continuous and φ belongs to O$_{M}$. Assumption 6 (generalized functions) . (b) The continuous functions w$_{·}$ belong to a bounded set in S $^{∗}$, S ∗ 0 $_{,m}$( V ); also, φ ′ k for k = 1 , ..., d, as well as φ − 1 and f − 1 x all belong to S ∗ 0 $_{,m}$( V ) . (c) The regression function g is absolutely integrable. (c’) The regression function g can be represented as a sum g = g$_{L}$$_{1}$ + g$_{g}$, where g$_{L}$$_{1}$ is absolutely integrable and g$_{g}$ is such that the Fourier transform of g$_{g}$ in S $^{∗}$, γ$_{g}$ = F t ( g$_{g}$ ) , is singular and thus has support set Λ$_{g}$ that is compact and of zero Lebesgue measure; Λ$_{g}$ is a proper subset of supp ( γ ) . 54