Takagi–Landsberg curve

The blancmange function is defined on the unit interval by

[math]\displaystyle{ {\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n}, }[/math]

where [math]\displaystyle{ s(x) }[/math] is defined by [math]\displaystyle{ s(x)=\min_{n\in{\bold Z}}|x-n| }[/math], that is, [math]\displaystyle{ s(x) }[/math] is the distance from x to the nearest integer. The infinite sum defining [math]\displaystyle{ blanc(x) }[/math] converges absolutely for all x, but the resulting curve is a fractal. The blancmange function is continuous but nowhere differentiable.

The Takagi–Landsberg curve is a slight generalization, given by

[math]\displaystyle{ T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x) }[/math]

for a parameter w; thus the blancmange curve is the case [math]\displaystyle{ w = 1 / 2 }[/math]. The value [math]\displaystyle{ H = − \log 2w }[/math] is known as the Hurst parameter. For [math]\displaystyle{ w = 1 / 4 }[/math], one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.