On perturbations of continuous maps

We give sufficient conditions for the following problem: given a topological space X, a metric space Y, a subspace Z of Y, and a continuous map f from X to Y, is it possible, by applying to f an arbitrarily small perturbation, to ensure that f(X) does not meet Z? We also give a relative variant: if f(X') does not meet Z for a certain subset X' of X, then we may keep f unchanged on X'. We also develop a variant for continuous sections of fibrations, and discuss some applications to matrix perturbation theory.