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Darboux's theorem (analysis) : ウィキペディア英語版
Darboux's theorem (analysis)
Darboux's theorem is a theorem in real analysis, named after Jean Gaston Darboux. It states that all functions that result from the differentiation of other functions have the intermediate value property: the image of an interval is also an interval.
When ''f'' is continuously differentiable (''f'' in ''C''1(())), this is a consequence of the intermediate value theorem. But even when ''f′'' is ''not'' continuous, Darboux's theorem places a severe restriction on what it can be.
==Darboux's theorem==
Let I be an open interval, f\colon I\to \R a real-valued differentiable function. Then f' has the intermediate value property: If a and b are points in I with a, then for every y between f'(a) and f'(b), there exists an x in () such that f'(x)=y.〔

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
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