Riesz-Nágy singular functions revisited

Jaume Paradís, Pelegrí Viader, Lluís Bibiloni

Research output: Contribution to journalArticleResearchpeer-review

29 Citations (Scopus)


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-Nágy family of functions and we relate it to a system for real number representation which we call (τ, τ - 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. © 2006 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)592-602
JournalJournal of Mathematical Analysis and Applications
Issue number1
Publication statusPublished - 1 May 2007


  • Metric number theory
  • Singular functions


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