Continuous selection for set-valued mappings

We study properties of set-valued mappings $F$ admitting a continuous selection $f$ which is a continuous $\epsilon$-selection (from the set of $\epsilon$-closest points) for the images $F(x)$ $(x\in X)$. This is interpreted as an $\epsilon$-selection for continuously varying sets in a space with continuously varying norms. We deduce new fixed-point theorems from the results obtained. We also study geometric-topological properties of sets all of whose $r$-neighbourhoods possess a continuous $\epsilon$-selection for every $\epsilon>0$. We obtain a characterization of such sets.