A singular function with a non-zero finite derivative

A singular function with a non-zero finite derivative

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 “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its g...

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Veröffentlicht in:Nonlinear analysis 2012-09, Vol.75 (13), p.5010-5014
Hauptverfasser: Fernández Sánchez, Juan, Viader, Pelegrí, Paradís, Jaume, Díaz Carrillo, Manuel
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Sprache:eng
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Construction
Derivatives
Mathematical analysis
Nonlinearity
Singular functions
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container_issue 13
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container_title Nonlinear analysis
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creator Fernández Sánchez, Juan
Viader, Pelegrí
Paradís, Jaume
Díaz Carrillo, Manuel
description 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 “classic” singular functions—Cantor’s, Hellinger’s, Minkowski’s, and the Riesz–Nágy one, including its generalizations and variants—the 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.
doi_str_mv 10.1016/j.na.2012.04.015
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subjects Construction
Derivatives
Mathematical analysis
Nonlinearity
Singular functions
title A singular function with a non-zero finite derivative
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