  The purpose of this work is threefold: (i) extend shadowing theory for
discontinuous and non-invertible systems, (ii) consider more general classes of
perturbations (for example, small only on average), (iii) establish a general
theory based on the property that the shadowing holds for the case of a single
perturbation. The "gluing" construction used in the analysis of the last
property turns out to be the key point of this theory.
