Difference between revisions of "Faber F1"

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(Created page with "The Faber functions are $$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|$$ and $$F_2(x)=\displaystyle\...")
 
 
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The Faber functions are
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The Faber function $F_1$ is defined by
$$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|$$
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$$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|.$$
and
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$$F_2(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{k!} \displaystyle\int_{m \in \mathbb{Z}} \left|2^{k!}x-m \right|.$$
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<div align="center">
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<gallery>
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File:Faberf1plot.png|Plot of $F_1$.
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</gallery>
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</div>
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=Properties=
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[[Faber F1 is continuous]]<br />
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[[Faber F1 is nowhere differentiable]]<br />
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=See Also=
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[[Faber F2]]
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=References=
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[https://pure.ltu.se/ws/files/30923977/LTU-EX-03320-SE.pdf]
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{{:Continuous nowhere differentiable functions footer}}
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[[Category:SpecialFunction]]

Latest revision as of 03:32, 6 July 2016

The Faber function $F_1$ is defined by $$F_1(x)=\displaystyle\sum_{k=1}^{\infty} \dfrac{1}{10^k} \displaystyle\inf_{m \in \mathbb{Z}} \left|2^{k!} x -m \right|.$$

Properties

Faber F1 is continuous
Faber F1 is nowhere differentiable

See Also

Faber F2

References

[1]

Continuous nowhere differentiable functions