| MSC: | 11A63 | Radix representation; digital problems {For metric results, see 11K16} |
| 39B22 | Equations for real functions {See also 26A51, 26B25} | |
| 26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |
Abstract:
This note is a supplement to the paper \cite{kru-kr2} on the partial
derivatives $T_n$ of de Rham's function $R_a(x)$ with respect to the
parameter $a$ at $a=1/2$. In particular, $T_0(x)=x$ and $T_1(x)=2T(x)$
where $T$ is Takagi's continuous nowhere differentiable function. We
present a new representation of $T_n$. From this we derive a limit relation
at dyadic rational points. Moreover, we show that real linear combinations
of $T_n$ with $n\ge1$ are nowhere differentiable. Thus we are able to prove
that the functions which appear e.g. in the well known formula of Coquet for
power sums of binary digital sums are nowhere differentiable. Finally, we
derive a corresponding formula for power sums of the number of zeros.