Difference between revisions of "Weierstrass nowhere differentiable function"
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The Weierstrass function is | The Weierstrass function is | ||
| − | $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^ | + | $$f(x)=\displaystyle\sum_{k=0}^{\infty} a^k \cos(b^k\pi x),$$ |
where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$. | where $0<a<1$ and $b \in \{1,3,5,7,9,\ldots\}$ such that $ab > 1+\dfrac{3}{2}\pi$. | ||
Revision as of 23:21, 21 January 2016
The Weierstrass function is $$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
Theorem: The Weierstrass function $f$ is continuous everywhere but differentiable nowhere.
Proof: █
