# Are there standard parameters for the Weierstrass nowhere differentiable function?

On Wikipedia, the Weierstass non-differentiable function is defined as: $$f(x)=\sum^{\infty}_{n=0}a^n\cos(b^n\pi x)$$

where $0<a<1$, $0<b$, and $ab>1+\frac 32 \pi$

Since it seems like, for any $a$ and $b$ that satisfy the above conditions, the resulting function will be continuous everywhere and differentiable nowhere, would it be more accurate to refer to the Weierstrass "function" as a family of functions? Are there a standard $a$ and $b$ that are used in discussing this function?

Are there a standard $a$ and $b$ that are used in discussing this function?

In my experientce, $a=1/2$ and $b=3$ are used most often, that is $$f(x)=\sum^{\infty}_{n=0}\frac{1}{2 ^n}\cos(3^n\pi x)$$ For example, these are the parameters are used on the illustration in Wikipedia article Weierstrass function.

Focusing on the more prominent local extrema on $[0,1]$, you will see repeated trisection of the interval, as in the construction of the standard Cantor set. This shows that $3^n$ was used inside the cosine. Also, the function evaluates to $2$ at $0$, implying $\sum_{n=0}^\infty a^n =2$, hence $a=1/2$.

Given that both trisection and scaling by $2^{-n}$ are very standard, I'll say that these are somehow canonical parameters.

### But they don't satisfy $ab>1+\frac32 \pi$

Very true. That's an ugly constraint anyway, imposed to make the proof easier. Originally Weierstrass put $ab>1$, but then, apparently inspecting the details of the proof, changed to the above. Still, $ab>1$ is the natural condition: vertical scaling beats horizontal. Hardy eventually proved that even $ab\ge 1$ suffices for nowhere differentiability.

Recommended reading: Continuous nowhere differentiable functions, Master's Thesis by Johan Thim.

### "Weierstrass functions" is fine

Just like Cantor sets are made using various scaling parameters, it makes sense to talk about Weierstrass functions.