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.