Singularity order of the Riesz-N{\'a}gy-Tak{\'a}cs function
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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 $g(a,p)= \frac{a\log p +(1-a)\log (1-p)}{a\log a+(1-a) \log (1-a)}\geq 1$.
Abstract
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 $g(a,p)= \frac{a\log p +(1-a)\log (1-p)}{a\log a+(1-a) \log (1-a)}\geq 1$.
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