I'm very interested in finding a way or hint for the construction of the Weierstrass function which is everywhere continuous but nowhere differentiable - let's call this (ECND). My most humble example is the iteration.
$y_0 = |1-x|$
$y_n = \frac{1}{2^n}|1-2^n y_{n-1}| \text{ ; for } n > 0$
The vertical lines are absolute values.
Iterating produces an serrated function over $(0,2)$ which seems to disappear. However, since I'm guessing this iteration keeps the length constant, this can't be a problem (i.e the function is still there and each iteration gives increases the serrations to twice the previous ones.). I'm not claiming it is ECND, but it is an idea.
Do you have any info on how the original function was produced? What inspired the definition? Is the cosine function the only appropriate option or are there other functions that can satisfy the ECND condition?
I leave some iterations here:
OK. This is quite the visual proof that the length is constant. I inverted the functions upside down. The iteration is
$y_0 = 1-|1-x|$
$y_n = \frac{1}{2^n}\left(1-|1-2^n y_{n-1}|\right) \text{ ; for } n > 0$