| MSC: | 26A27 | Nondifferentiability (nondifferentiable functions, points of nondifferentiability), discontinuous derivatives |
| 39B22 | Equations for real functions [See also 26A51, 26B25] |
Abstract:
In this paper we derive functional
relations for Takagi's continuous nowhere differentiable function
$T$, and we give an explicit representation of $T$ at dyadic points.
As application of these functional relations we derive a limit
relation at dyadic points which implies that at these points $T$
attains locally minima. Further, $T$ is maximal on a perfect set of
Lebesgue measure zero. Though the points, where $T$ has a locally
maximum, are dense it is remarkable that there is no point where $T$
has a {\it proper} maximum. Moreover, we verify the existence of the
improper derivatives $T'(x)=+\infty$ or $T'(x)=-\infty$ for rational
$x$ which have an odd length of period in the binary representation.
Finally we investigate
one-side upper and lower derivatives.