A singular function with a non-zero finite derivative

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• 作者：Juan Fern&aacute ; ndez S&aacute ; nchez^a ; ^{juanfernandez@ual.es} ; Pelegrí ; Viader^b ; ^{pelegri.viader@upf.edu} ; Jaume Paradí ; s^b ; ^{jaume.paradis@upf.edu} ; Manuel Dí ; az Carrillo^c ; ^{madiaz@ugr.es}
• 关键词：Singular functions
• 刊名：Nonlinear Analysis
• 出版年：2012
• 期刊代码：43_0362546x
• 类别：mt
• 出版时间：September, 2012
• 卷：75
• 期：13
• 页码：5010-5014
• 文件大小：234 K

摘要

This paper exhibits, for the first time in the literature, a continuous strictly increasing singular function with a derivative that takes non-zero finite values at some points. For all the known 鈥渃lassic鈥?singular functions鈥擟antor鈥檚, Hellinger鈥檚, Minkowski鈥檚, and the Riesz-N谩gy one, including its generalizations and variants鈥攖he derivative, when it existed and was finite, had to be zero. As a result, there arose a strong suspicion (almost a conjecture) that this had to be the case for any singular function. We present here a singular function, constructed as a patchwork of known classic singular functions, with derivative 1 on a subset of the Cantor set.