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In mathematics, Volterra's function, named for Vito Volterra, is a real-valued function ''V'' defined on the real line R with the following curious combination of properties: * ''V'' is differentiable everywhere * The derivative ''V'' ′ is bounded everywhere * The derivative is not Riemann-integrable. ==Definition and construction== The function is defined by making use of the Smith–Volterra–Cantor set and "copies" of the function defined by for ''x'' ≠ 0 and ''f''(''x'') = 0 for ''x'' = 0. The construction of ''V'' begins by determining the largest value of ''x'' in the interval (1/8 ) for which ''f'' ′(''x'') = 0. Once this value (say ''x''0) is determined, extend the function to the right with a constant value of ''f''(''x''0) up to and including the point 1/8. Once this is done, a mirror image of the function can be created starting at the point 1/4 and extending downward towards 0. This function will be defined to be 0 outside of the interval (1/4 ). We then translate this function to the interval (5/8 ) so that the resulting function, which we call ''f''1, is nonzero only on the middle interval of the complement of the Smith–Volterra–Cantor set. To construct ''f''2, ''f'' ′ is then considered on the smaller interval (), truncated at the last place the derivative is zero, extended, and mirrored the same way as before, and two translated copies of the resulting function are added to ''f''1 to produce the function ''f''2. Volterra's function then results by repeating this procedure for every interval removed in the construction of the Smith–Volterra–Cantor set; in other words, the function ''V'' is the limit of the sequence of functions ''f''1, ''f''2, ... 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「Volterra's function」の詳細全文を読む スポンサード リンク
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