Difference between revisions of "Takagi function"

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Define $s(x)=\inf_{n \in \mathbb{Z}} |x-n|$ (i.e. the distance from $x$ to the set of integers $\mathbb{Z}$). The Takagi function (also called the blancmange function) is defined by
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The Takagi function (also called the blancmange function) is defined by
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$$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
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$$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n}.$$
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Note: to calculate $s(x)$ you may use $s(x)=\min \left(2^n x - \lfloor 2^n x \rfloor, \lceil 2^n x \rceil - x \right)$, where $\lfloor \cdot \rfloor$ denotes the [[floor]] function and $\lceil \cdot \rceil$ denotes the [[ceiling]] function.
 
 
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Revision as of 03:13, 6 July 2016

The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n}.$$

  • Graph of $\mathrm{takagi}$ on $[0,1]$.

Properties

Theorem: The Takagi function is continuous on $\mathbb{R}$.

Proof: █

Theorem: The Takagi function is nowhere differentiable on $\mathbb{R}$.

Proof: █

See Also

van der Waerden function

References

[1]
[2]

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