# Differentiability of the Weierstrauss Function

• Jun 15th 2011, 02:40 PM
Also sprach Zarathustra
Differentiability of the Weierstrass Function
I want to know something about:

$\mathfrak{ Weierstrass \ function:}$

$\sum_{n=0}^{\infty}a^n\cos(b^n\pi x)$ where $0 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.
• Jun 16th 2011, 06:58 PM
ojones
Re: Differentiability of the Weierstrauss Function
For (1), see Weierstrass's paper. For (2), these conditions are artificial and unnecessary. See Hardy's paper on a generalization.