Differentiability of the Weierstrauss Function

**Also sprach Zarathustra** · June 15th 2011, 02:40 PM

I want to know something about:

$\mathfrak{ Weierstrass \ function:}$

$\sum_{n=0}^{\infty}a^n\cos(b^n\pi x)$ where $0<a<1$ and $b$ is odd integer so $ab>1+\frac{3}{2}\pi$ .

1. How to prove that Weierstrass function is nowhere differentiable?

2. Why $b$ must be odd? And why $ab>1+\frac{3}{2}\pi$ ?

Thank you.

**ojones** · June 16th 2011, 06:58 PM

For (1), see Weierstrass's paper. For (2), these conditions are artificial and unnecessary. See Hardy's paper on a generalization.

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