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In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is also referred to as the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor-Vitali function, the Devil's staircase,〔.〕 the Cantor staircase function,〔http://mathworld.wolfram.com/CantorStaircaseFunction.html〕 and the Cantor-Lebesgue function.〔.〕 introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack. The Cantor function was discussed and popularized by , and . ==Definition== See figure. To formally define the Cantor function ''c'' : () → (), let ''x'' be in () and obtain ''c''(''x'') by the following steps: #Express ''x'' in base 3. #If ''x'' contains a 1, replace every digit after the first 1 by 0. #Replace all 2s with 1s. #Interpret the result as a binary number. The result is ''c''(''x''). For example: * 1/4 becomes 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... When read in base 2, this corresponds to 1/3 in base 10, so ''c''(1/4) = 1/3. * 1/5 becomes 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since there are no 2s. When read in base 2, this corresponds to 1/4 in base 10, so ''c''(1/5) = 1/4. * 200/243 becomes 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. When read in base 2, this corresponds to 3/4 in base 10, so ''c''(200/243) = 3/4. 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「Cantor function」の詳細全文を読む スポンサード リンク
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