Difference between revisions of "Weierstrass nowhere differentiable function"

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The Weierstrass function is
 
The Weierstrass function is
$$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^n\pi x),$$
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$$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^k\pi x),$$
 
where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.
 
where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed">
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[[Weierstrass function is continuous]]<br />
<strong>Theorem:</strong> The Weierstrass function $f$ is [[continuous]] everywhere but [[derivative|differentiable]] nowhere.
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[[Weierstrass function is nowhere differentiable]]<br />
<div class="mw-collapsible-content">
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<strong>Proof:</strong>
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=Videos=
</div>
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[https://www.youtube.com/watch?v=pCEFZk9Vihs Weierstrass example (17 October 2013)]<br />
</div>
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=References=
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[http://kryakin.org/at/hardy_1916_W.pdf]
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[[Category:SpecialFunction]]

Latest revision as of 17:54, 25 June 2017

The Weierstrass function is $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^k\pi x),$$ where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.

Properties

Weierstrass function is continuous
Weierstrass function is nowhere differentiable

Videos

Weierstrass example (17 October 2013)

References

[1]