On properties of continuous mapings of nonlimited metric spaces
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 i...
Збережено в:
| Дата: | 1991 |
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| Автори: | , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1991
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/9626 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | 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 $F$. |
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