Difference between revisions of "Weierstrass nowhere differentiable function"
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(Created page with "The Weierstrass function is $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^n\pi x),$$ where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$. =...") |
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| − | <strong>Theorem:</strong> The Weierstrass function $f$ is [[continuous]] everywhere but [[differentiable]] nowhere. | + | <strong>Theorem:</strong> The Weierstrass function $f$ is [[continuous]] everywhere but [[derivative|differentiable]] nowhere. |
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<strong>Proof:</strong> █ | <strong>Proof:</strong> █ | ||
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</div> | </div> | ||
Revision as of 20:39, 21 May 2015
The Weierstrass function is $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^n\pi x),$$ where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$.
Properties
Theorem: The Weierstrass function $f$ is continuous everywhere but differentiable nowhere.
Proof: █
