Continuous Selections as Uniform Limits of δ-Continuous ε-Selections

We prove a selection theorem which unifies the nonconvex-valued and the zero-dimensional selection theorems. Our proof is based on a new method which unifies the methods of outside and inside approximations of a given lower semicontinuous multivalued mapping. As an application, we obtain a unified selection theorem ('modulo" a countable subset of the domain) and a theorem on density of selections.