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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 Monotonic function ： ウィキペディア英語版 Monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. == Monotonicity in calculus and analysis == In calculus, a function $f$ defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely increasing or decreasing. It is called monotonically increasing (also increasingor nondecreasing), if for all $x$ and $y$ such that $x\; \backslash leq\; y$ one has $f\backslash !\backslash left(x\backslash right)\; \backslash leq\; f\backslash !\backslash left(y\backslash right)$, so $f$ preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasingor nonincreasing) if, whenever $x\; \backslash leq\; y$, then $f\backslash !\backslash left(x\backslash right)\; \backslash geq\; f\backslash !\backslash left(y\backslash right)$, so it ''reverses'' the order (see Figure 2). If the order $\backslash leq$ in the definition of monotonicity is replaced by the strict order , then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for $x$ not equal to $y$, either or $x\; >\; y$ and so, by monotonicity, either or $f\backslash !\backslash left(x\backslash right)\; >\; f\backslash !\backslash left(y\backslash right)$, thus $f\backslash !\backslash left(x\backslash right)$ is not equal to $f\backslash !\backslash left(y\backslash right)$.) When functions between discrete sets are considered in combinatorics, it is not always obvious that "increasing" and "decreasing" are taken to include the possibility of repeating the same value at successive arguments, so one finds the terms weakly increasing and weakly decreasing to stress this possibility. The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the function of figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it is neither non-decreasing nor non-increasing. The term monotonic transformation can also possibly cause some confusion because it refers to a transformation by a strictly increasing function. Notably, this is the case in economics with respect to the ordinal properties of a utility function being preserved across a monotonic transform (see also monotone preferences).〔See the section on Cardinal Versus Ordinal Utility in .〕 A function $f\backslash !\backslash left(x\backslash right)$ is said to be absolutely monotonic over an interval $\backslash left(a,\; b\backslash right)$ if the derivatives of all orders of $f$ are nonnegative at all points on the interval. 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「Monotonic function」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.