# Thread: Differentiability of the Weierstrauss Function

1. ## Differentiability of the Weierstrass Function

I want to know something about:

$\displaystyle \mathfrak{ Weierstrass \ function:}$

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

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

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

Thank you.

2. ## 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.