is considered without requiring differentiability; it is shown that for this case the conditional distribution and conditional density have a straightforward representation as generalized functions on appropriate spaces. The repre- sentation is in terms of functionals involving the conditioning distribution (rather than the conditioning variable) as an argument; this representation avoids the nonlinearity introduced by the denominator. When the usual rep- resentation holds, a simple correspondence between the two representations is established. Conditional density, f$_{y}$$_{|}$$_{x}$ is defiend as a generalized derivative of the conditional distribution generalized function. 

The convergence of the usual kernel estimator of the conditional distribu- tion is known under smoothness assumptions (Pagan and Ullah, 1999, Li and Racine, 2007) and utilizes the properties of the kernel density estimator; the density appears in the denominator of the statistic requiring some support assumptions and possibly regularization to converge. Here the root-n con- vergence of the kernel estimator to a limit Gaussian process in generalized function space is established without any extra restrictions on the distribu- tion. 

An interpretation of a conditional moment function is provided here in the space of generalized functions, thus again without any restriction beyond continuity of conditioning distribution. For estimators, such as for condi- tional mean kernel estimator the asymptotic properties are established, the result is then that root-n convergence in generalized functions obtains for the kernel estimator without any restrictions on smoothness of distribution 

