Blancmange function - a continuous, nowhere differentiable function.

Blancmange function is the sum of the infinite series of sawtooth functions, each with half the height and half the wavelength of the previous one.


f(x) =  

+∞
Σ
k=0


2-kg(2kx)


            (1)
where: g(x) = min( x-[x], [x]+1-x )

This function is introduced in 1930 by Bartel Leendert Van der Waerden (1903-1996).


PROPERTIES:
  1. Blancmange function is continuos, since g(x) is continous and the series is uniformly convergent.
  2. The nth remainder Rn(x) of series (1) is periodic with period 2-n.
  3. Blancmange function is nowhere differentiable.PROOF
  4. The nth remainder Rn(x) of series (1) is the scaled down transform of f(x) with coeficient 2-n. (e.g. f(x)=2nRn(2-nx) and 2nRn(x)=f(2nx))View
Click for a larger graph. | Another graphing.
While Blancmange curve (the graph of Blancmange function) cannot be drawn, the curve shown is a "best effort" presentation of the Blancmange curve (for x nonnegative).

Blancmange function is nowhere differentiable.