Weierstrass function
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- The Weierstrass function may also refer to the Weierstrass elliptic function (<math>\wp</math>) or the Weierstrass sigma, zeta or eta functions.
In mathematics, the Weierstrass function was the first example found of a kind of function with the property that it is continuous everywhere but differentiable nowhere. Weierstrass functions are defined by
- <math>f(x)=\sum_{n=0}^\infty a^n\cos(b^n\pi x),</math>
where <math>0<a<1</math> and
- <math> ab>1+\frac{3}{2}\pi.</math>
What makes this function significant is that, similar to a fractal, it has uniform and infinite complexity no matter how closely one "zooms in" to view the image. For this reason, curves do not appear to linearize as one "zooms in"; thus no tangent can be equated to the graph at any one point. Hence the function is not differentiable.
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See also
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