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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 Froda's theorem ： ウィキペディア英語版 Froda's theorem In mathematics, Darboux-Froda's theorem, named after Alexandru Froda, a Romanian mathematician, describes the set of discontinuities of a (monotone) real-valued function of a real variable. Usually, this theorem appears in literature without a name. It was written in A. Froda' thesis in 1929 .〔Alexandru Froda, (Sur la Distribution des Propriétés de Voisinage des Fonctions de Variables Réelles ), These, Harmann, Paris, 3 December 1929〕〔''Alexandru Froda – Collected Papers (Opera Matematica), Vol.1'', Ed. Academ. Romane, 2000〕. As it is acknowledged in the thesis, it is in fact due Jean Gaston Darboux 〔 Jean Gaston Darboux (Mémoire sur les fonctions discontinues ), Annales de l'École Normale supérieure, 2-ème série, t. IV, 1875, Chap VI. 〕 ==Definitions== #Consider a function of real variable with real values defined in a neighborhood of a point $x\_0$ and the function is discontinuous at the point on the real axis $x\; =\; x\_0$. We will call a removable discontinuity or a jump discontinuity a discontinuity of the first kind.〔Walter Rudin, ''Principles of Mathematical Analysis'', McGraw-Hill 1964, (Def. 4.26, pp. 81–82)〕 #Denote $f(x+0):=\backslash lim\_f(x+h)$ and $f(x-0):=\backslash lim\_f(x-h)$. Then if $f(x\_0+0)$ and $f(x\_0-0)$ are finite we will call the difference $f(x\_0+0)-f(x\_0-0)$ the jump〔M. Nicolescu, N. Dinculeanu, S. Marcus, ''Mathematical Anlaysis'' (Bucharest 1971), Vol.1, Pg.213, (Romanian )〕 of f at $x\_0$. If the function is continuous at $x\_0$ then the jump at $x\_0$ is zero. Moreover, if $f$ is not continuous at $x\_0$, the jump can be zero at $x\_0$ if $f(x\_0+0)=f(x\_0-0)\backslash neq\; f(x\_0)$. 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「Froda's 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.