Takagi–Landsberg curve

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The blancmange function is defined on the unit interval by

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

where [math]s(x)[/math] is defined by [math]s(x)=\min_{n\in{\bold Z}}|x-n|[/math], that is, [math]s(x)[/math] is the distance from x to the nearest integer. The infinite sum defining [math]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] 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]w = 1 / 2[/math]. The value [math]H = − \log2w[/math] is known as the Hurst parameter. For [math]w = 1 / 4[/math], one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.