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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 Semi-continuity ： ウィキペディア英語版 Semi-continuity :''For the notion of upper or lower semicontinuous multivalued function see: Hemicontinuity'' In mathematical analysis, semi-continuity (or semicontinuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function ''f'' is upper (respectively, lower) semi-continuous at a point ''x''0 if, roughly speaking, the function values for arguments near ''x''0 are either close to ''f''(''x''0) or less than (respectively, greater than) ''f''(''x''0). == Examples == Consider the function ''f'', piecewise defined by ''f''(''x'') = –1 for ''x'' < 0 and ''f''(''x'') = 1 for ''x'' ≥ 0. This function is upper semi-continuous at ''x''0 = 0, but not lower semi-continuous. The indicator function of an open set is lower semi-continuous, whereas the indicator function of a closed set is upper semi-continuous. The floor function $f(x)=\backslash lfloor\; x\; \backslash rfloor$, which returns the greatest integer less than or equal to a given real number ''x'', is everywhere upper semi-continuous. Similarly, the ceiling function $f(x)=\backslash lceil\; x\; \backslash rceil$ is lower semi-continuous. A function may be upper or lower semi-continuous without being either left or right continuous. For example, the function :$f(x)\; =\; \backslash begin$ 1 & \mbox x < 1,\\ 2 & \mbox x = 1,\\ 1/2 & \mbox x > 1, \end is upper semi-continuous at ''x'' = 1 although not left or right continuous. The limit from the left is equal to 1 and the limit from the right is equal to 1/2, both of which are different from the function value of 2. Similarly the function :$f(x)\; =\; \backslash begin$ \sin(1/x) & \mbox x \neq 0,\\ 1 & \mbox x = 0, \end is upper semi-continuous at ''x'' = 0 while the function limits from the left or right at zero do not even exist. If $X=\backslash mathbb\; R^n$ is a Euclidean space (or more generally, a metric space) and $\backslash Gamma=C((),X)$ is the space of curves in $X$ (with the supremum distance $d\_\backslash Gamma(\backslash alpha,\backslash beta)=\backslash sup\_t\backslash \; d\_X(\backslash alpha(t),\backslash beta(t))$, then the length functional $L:\backslash Gamma\backslash to()$, which assigns to each curve $\backslash alpha$ its length $L(\backslash alpha)$, is lower semicontinuous. Let $(X,\backslash mu)$ be a measure space and let $L^+(X,\backslash mu)$ denote the set of positive measurable functions endowed with the topology of convergence in measure with respect to $\backslash mu$. Then the integral, seen as an operator from $L^+(X,\backslash mu)$ to $()$ is lower semi-continuous. This is just Fatou's lemma. 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「Semi-continuity」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.