On bounding, interior and covering functions

AbstractWe prove three theorems yielding sufficient conditions for a continuous function f: X → Y to have no isolated bounding points (i.e., points at which it is not an open mapping), to be interior a z (i.e.,f(z) ϵ f(X)0), and to have its image cover a segment. In the first theorem X and Y are rather general topological spaces and some applications to optimal control are discussed. In the second and third theorems X an Y are finite-dimensional and we use the concepts of ordinary and unbounded derivate containers which are a form of set-valued derivatives. The proof of the second theorem also involves the topological degree of f