Difference between revisions of "Takagi function"

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Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Takagi function (also called the blancmange function) is defined by
 
Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Takagi function (also called the blancmange function) is defined by
 
$$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
 
$$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$
 
+
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 17:25, 22 January 2016

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

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]