Takagi function

From specialfunctionswiki
Revision as of 17:27, 22 January 2016 by Tom (talk | contribs)
Jump to: navigation, search

Define the function $s(x)=\inf_{n \in \mathbb{Z}} |x-n|$ (i.e. the distance from $x$ to the set of integers $\mathbb{Z}$). The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$ 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]