We consider the inverse regression model Y = Hf (X) + ε for several classes of nonlinear Hammerstein integral operators H. In particular identifiability depending on the integral kernel is discussed. We introduce estimators for parametric functions f with discontinuities of certain order including piecewise polynomials with kinks or jumps or free-knot splines respectively. We derive rates of convergence and asymptotic normality of these estimators and a data example from rheology illustrates

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... results. An extension of the model for functions f from approximation spaces of parametric piecewise continuous functions is presented. 7 Approximation spaces 7.1 Definition .