下半Cauchy列的研究
A study on downward half Cauchy sequences
摘要：
本文引入并研究了下连续性和下紧性的概念。$\R$的子集$E$上的实值函数$f$，如果它保持向下半柯西序列，则实数集是向下连续的，即序列$(f(α_{n}))$是向下半柯西的，只要$(α_{n})$是$E$中的点的向下半柯西序列，其中，如果对于每个$\R$，存在一个$n_{0}\in{\N}$，使得$α_{m}-α_{n} <\varepsilon$用于$m \geq n \geq n_0$。证明了下连续函数集是连续函数集的真子集。
英文摘要：
In this paper, we introduce and investigate the concepts of down continuity and down compactness. A real valued function $f$ on a subset $E$ of $\R$, the set of real numbers is down continuous if it preserves downward half Cauchy sequences, i.e. the sequence $(f(α_{n}))$ is downward half Cauchy whenever $(α_{n})$ is a downward half Cauchy sequence of points in $E$, where a sequence $(α_{ k})$ of points in $\R$ is called downward half Cauchy if for every $\varepsilon>0$ there exists an $n_{0}\in{\N}$ such that $α_{m}-α_{n} <\varepsilon$ for $m \geq n \geq n_0$. It turns out that the set of down continuous functions is a proper subset of the set of continuous functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
