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