Difference between revisions of "Takagi function"
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(Created page with "Define the function $s(x)=\min_{n \in \mathbb{Z}} |x-n|$. The Blancmange function is defined by $$\mathrm{blanc}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{s(2^n x)}{2^n}.$$") |
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| − | + | The Takagi function (also called the blancmange function) is defined by | |
| − | $$\mathrm{ | + | $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n},$$ |
| + | where $\mathrm{dist}_{\mathbb{Z}}$ denotes the [[distance to integers]] function. | ||
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Takagiplot.png|Graph of $\mathrm{takagi}$ on $[0,1]$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | [[Takagi function is continuous]]<br /> | ||
| + | [[Takagi function is nowhere differentiable]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[van der Waerden function]] | ||
| + | |||
| + | =References= | ||
| + | [https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]<br /> | ||
| + | [http://www.math.tamu.edu/~tvogel/gallery/node7.html]<br /> | ||
| + | |||
| + | {{:Continuous nowhere differentiable functions footer}} | ||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:33, 6 July 2016
The Takagi function (also called the blancmange function) is defined by $$\mathrm{takagi}(x)=\displaystyle\sum_{k=0}^{\infty} \dfrac{\mathrm{dist}_{\mathbb{Z}}(2^n x)}{2^n},$$ where $\mathrm{dist}_{\mathbb{Z}}$ denotes the distance to integers function.
Graph of $\mathrm{takagi}$ on $[0,1]$.
Properties
Takagi function is continuous
Takagi function is nowhere differentiable
