Add to ORKGBy the Shadowing Lemma we can shadow any sufficient accurate pseudo-trajectory of a hyperbolic system by a true trajectory of a hyperbolic system. If we are interested in finite trajectories, at least from one side, then a pseudo trajectory usually has many possible shadows. Here we show that we can choose a continuous single-valued selector from the corresponding multi-valued operator 'pseudo-trajectory#->#the totality of possible shadows'. We do this in the context of Lipschitz mappings which are semi-hyperbolic on some compact subset, which need not be invariant. We also prove that semi-hyperbolicity implis inverse shadowing with respect to a very broad class of nonsmooth perturbations. (orig.)Available from TIB Hannover: RR 5549(346)...