Riesz–Nágy singular functions revisited. (English) Zbl 1115.26001
The authors revisit the Riesz-Nágy family of strictly increasing singular functions built as the limit of a sequence of deformations of the identity function, and the further result by Takács, where a closed form definition of a family of increasing continuous singular functions was provided. In particular, both families are related to a system for real number representation called \((\tau, \tau-1)\)-expansion: given \(\tau\in {\mathbb R},\) \(\tau>1,\) any \(x\in [0,1)\) can be expressed as \(x=\sum_1^{\infty} \varepsilon_i{1\over \tau^i}(\tau -1)^{\sum_{j=1}^{i-1}\varepsilon_j},\) with \(\varepsilon_i\in \{0,1\}.\) After a detailed study of these expansions, exploiting their metrical properties, they prove the singularity of the family. Moreover, with the help of \((\tau,\tau-1)\)-expansions, they extend the family and obtain conditions for the null and infinite derivatives.
Reviewer: Rita Pini (Milano)
MSC:
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
| 26A48 | Monotonic functions, generalizations |
Keywords:
metric number theoryReferences:
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