On properties of continuous mapings of nonlimited metric spaces

Abstract

Suppose a closed unbounded set $F\subset R_n$ 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 $\infty$, at which it is assumed that $f$ has a finite limit $A_i\in R^{(2)}, i=1,2,\dots,p$.

It is proved that: 1) $f$  is bounded on $F$; 2) if $f$ is a real functional, then the set $f(F)U (\bigcup_{i=1}^pA_i)$ contains a smallest and a largest value; 3) if the distance between $F_i$  and $F_j$ is greater than zero whenever $i\ne j$, then $f$  is uniformly continuous on $F$.