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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Heine–Cantor theorem ： ウィキペディア英語版 Heine–Cantor theorem In mathematics, the Heine–Cantor theorem, named after Eduard Heine and Georg Cantor, states that if ''f'' : ''M'' → ''N'' is a continuous function between two metric spaces, and ''M'' is compact, then ''f'' is uniformly continuous. An important special case is that every continuous function from a closed interval to the real numbers is uniformly continuous. ==Proof== Suppose that ''M'' and ''N'' are two metric spaces with metrics ''dM'' and ''dN'', respectively. Suppose further that $f:\; M\; \backslash to\; N$ is continuous, and that ''M'' is compact. We want to show that ''f'' is uniformly continuous, that is, for every $\backslash epsilon\; >\; 0$ there exists $\backslash delta\; >\; 0$ such that for all points ''x'',''y'' in the domain ''M'', implies that . Fix some positive $\backslash epsilon\; >\; 0$. Then by continuity, for any point ''x'' in our domain ''M'', there exists a positive real number $\backslash delta\_x\; >\; 0$ such that when ''y'' is within $\backslash delta\_x$ of ''x''. Let ''Ux'' be the open $\backslash delta\_x/2$-neighborhood of ''x'', i.e. the set :$U\_x\; =\; \backslash left\backslash \backslash delta\_x\; \backslash right\backslash \}$ Since each point ''x'' is contained in its own ''Ux'', we find that the collection $\backslash$ is an open cover of ''M''. Since ''M'' is compact, this cover has a finite subcover. That subcover must be of the form :$U\_,\; U\_,\; \backslash ldots,\; U\_$ for some finite set of points $\backslash \; \backslash subset\; M$. Each of these open sets has an associated radius $\backslash delta\_/2$. Let us now define $\backslash delta\; =\; \backslash min\_\; \backslash frac\backslash delta\_$, i.e. the minimum radius of these open sets. Since we have a finite number of positive radii, this number $\backslash delta$ is well-defined. We may now show that this $\backslash delta$ works for the definition of uniform continuity. Suppose that for any two ''x,y'' in ''M''. Since the sets $U\_$ form an open (sub)cover of our space ''M'', we know that ''x'' must lie within one of them, say $U\_$. Then we have that . The Triangle Inequality then implies that : implying that ''x'' and ''y'' are both at most $\backslash delta\_$ away from ''xi''. By definition of $\backslash delta\_$, this implies that $d\_N(f(x\_i),f(x))$ and $d\_N(f(x\_i),\; f(y))$ are both less than $\backslash epsilon/2$. Applying the Triangle Inequality then yields the desired : For an alternative proof in the case of ''M'' = (''b'' ) a closed interval, see the article on non-standard calculus. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Heine–Cantor theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.