# Takagi–Landsberg curve

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The blancmange function is defined on the unit interval by

${\rm blanc}(x) = \sum_{n=0}^\infty {s(2^{n}x)\over 2^n},$

where $s(x)$ is defined by $s(x)=\min_{n\in{\bold Z}}|x-n|$, that is, $s(x)$ is the distance from x to the nearest integer. The infinite sum defining $blanc(x)$ 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

$T_w(x) = \sum_{n=0}^\infty w^n s(2^{n}x)$

for a parameter w; thus the blancmange curve is the case $w = 1 / 2$. For $w = 1 / 4$, one obtains the parabola: the construction of the parabola by midpoint subdivision was described by Archimedes.

### The JavaScript code to produce this picture

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### References

• Teiji Takagi, "A Simple Example of a Continuous Function without Derivative", Proc. Phys. Math. Japan, (1903) Vol. 1, pp. 176-177.