Properties of continuous mappings of unbounded metric spaces

Abstract

Suppose a closed unbounded set F ⊂R_n is a union of a finite number p of closed unbounded sets F_i that are pairwise disjoint, and suppose f is a continuous mapping of F into the metric space R⁽²⁾. With each set F_i there is associated a point at infinity ∞_i, at which it is assumed that f has a finite limit A_i ε R⁽²⁾, i=1, 2, ..., p. It is proved that: 1) f is bounded on F; 2) if f is a real functional, then the set$f(F) \cup \left( {\bigcup\limits_{i = 1}^p {A_i } } \right)$ contains a smallest and a largest value; 3) if the distance between F_i and F_j is greater than zero whenever i ≠ j, then f is uniformly continuous on F.