# Takagi–Landsberg curve

The blancmange function is defined on the unit interval by

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

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

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

for a parameter w; thus the blancmange curve is the case $\displaystyle{ w = 1 / 2 }$. For $\displaystyle{ w = 1 / 4 }$, 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.