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Riesz-Nágy singular functions revisited

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Abstract

In 1952 F. Riesz and Sz.Nágy published an example of a monotonic continuous function whose derivative is zero almost everywhere, that is to say, a singular function. Besides, the function was strictly increasing. Their example was built as the limit of a sequence of deformations of the identity function. As an easy consequence of the definition, the derivative, when it existed and was finite, was found to be zero. In this paper we revisit the Riesz-N´agy family of functions and we relate it to a system for real number representation which we call (t, t-1)–expansions. With the help of these real number expansions we generalize the family. The singularity of the functions is proved through some metrical properties of the expansions used in their definition which also allows us to give a more precise way of determining when the derivative is 0 or infinity.

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  • Jaume Paradís & Pelegrí Viader & Lluís Bibiloni, 2006. "Riesz-Nágy singular functions revisited," Economics Working Papers 941, Department of Economics and Business, Universitat Pompeu Fabra.
    • Handle: RePEc:upf:upfgen:941
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    Keywords

    Singular functions; metric number theory;

    JEL classification:

    • C00 - Mathematical and Quantitative Methods - - General - - - General

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