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 $f\left( {{X}^{'}} \right)$ does not meet $Z$ ? We also give a relative variant: if $f\left( X\prime\right)$ does not meet $Z$ for a certain subset ${X}'\subset X$ , 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.