Commun. Korean Math. Soc. 2015; 30(1): 7-21
Printed January 31, 2015
https://doi.org/10.4134/CKMS.2015.30.1.7
Copyright © The Korean Mathematical Society.
In-Soo Baek
Busan University of Foreign Studies
We give the characterization of H\"older differentiability points and non-differentiability points of the Riesz-N{\'a}gy-Tak{\'a}cs (RNT) singular function $\Psi_{a,p}$ satisfying $\Psi_{a,p}(a)=p$. It generalizes recent multifractal and metric number theoretical results associated with the RNT function. Besides, we classify the singular functions using the singularity order deduced from the H\"older derivative giving the information that a strictly increasing smooth function having a positive derivative Lebesgue almost everywhere has the singularity order 1 and the RNT function $\Psi_{a,p}$ has the singularity order $g(a,p)= \frac{a\log p +(1-a)\log (1-p)}{a\log a+(1-a) \log (1-a)}\geq 1$.
Keywords: Hausdorff dimension, packing dimension, distribution set, local dimension set, singular function, metric number theory, H\"older derivative
MSC numbers: Primary 28A78; Secondary 26A30
2015; 30(3): 221-226
2005; 20(4): 695-702
2007; 22(4): 547-552
2008; 23(4): 549-554
© 2022. The Korean Mathematical Society. Powered by INFOrang Co., Ltd