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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 Pompeiu derivative ： ウィキペディア英語版 Pompeiu derivative In mathematical analysis, a Pompeiu derivative is a real-valued function of one real variable that is the derivative of an everywhere differentiable function and that vanishes in a dense set. In particular, a Pompeiu derivative is discontinuous at any point where it is not 0. Whether non-identically zero such functions may exist was a problem that arose in the context of early-1900s research on functional differentiability and integrability. The question was affirmatively answered by Dimitrie Pompeiu by constructing an explicit example; these functions are therefore named after him. ==Pompeiu's construction== Pompeiu's construction is described here. Let $\backslash sqrt()$ denote the real cubic root of the real number $x.$ Let $\backslash \_$ be an enumeration of the rational numbers in the unit interval $().$ Let $\backslash \_$ be positive real numbers with Define, for all $x\backslash in\; ()$ :$g(x):=\backslash sum\_^\backslash infty\; \backslash ,a\_j\; \backslash sqrt().$ Since for any $x\backslash in()$ each term of the series is less than or equal to ''a''j in absolute value, the series uniformly converges to a continuous, strictly increasing function g(x), due to the Weierstrass M-test. Moreover, it turns out that the function ''g'' is differentiable, with :$g^(x):=\backslash frac\backslash sum\_^\backslash infty\; \backslash frac(x):=+\backslash infty.$ Since the image of $g$ is a closed bounded interval with left endpoint $0=g(0),$ up to a multiplicative constant factor one can assume that ''g'' maps the interval $()$ onto itself. Since ''g'' is strictly increasing, it is a homeomorphism; and by the theorem of differentiation of the inverse function, its composition inverse $f\backslash ,:=g^$ has a finite derivative at any point, which vanishes at least in the points $\backslash \_.$ These form a dense subset of $()$ (actually, it vanishes in many other points; see below). 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「Pompeiu derivative」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.