Bolzano function

Bernard Bolzano

The Bolzanofunktion is historically the first construction of a function which, although steadily , but nowhere differentiable is. It is named after its discoverer Bernard Bolzano , it was found by him around 1830 and presented in his manuscript Functionenlehre (which was not published until 1930).

The possibility of the existence of continuous but nowhere differentiable functions became known through Karl Weierstrass (lecture at the Berlin Academy in 1872), which at the time had a shocking effect on many. Weierstrass' example function was published by Paul Du Bois-Reymond in 1875. Also Bernhard Riemann presented such in 1861 in his lectures, and since then many more have been constructed.

definition

The Bolzano function is defined as the limit value of a function sequence. Furthermore, one can select the domain and the image set as arbitrary closed intervals of real numbers.

So be the desired domain and the desired amount of images. ${\displaystyle [a,b]}$ ${\displaystyle [A,B]}$

Transformation of a linear piece from (dashed) to a component of (solid) ${\displaystyle B_{k-1}}$ ${\displaystyle B_{k}}$

${\displaystyle B_{1}}$is a linear function with vertices , defined as: ${\displaystyle B_{1}(a)=A}$ ${\displaystyle B_{1}(b)=B}$

${\displaystyle B_{1}(x):=A+{\frac {B-A}{b-a}}(x-a)}$

${\displaystyle B_{2}}$ is defined as a piecewise linear function on four intervals, with the following five corner points:

1. ${\displaystyle B_{2}(a)=A}$
2. ${\displaystyle B_{2}\left(a+{\frac {3}{8}}(b-a)\right)=A+{\frac {5}{8}}(B-A)}$
3. ${\displaystyle B_{2}\left({\frac {1}{2}}(a+b)\right)=A+{\frac {1}{2}}(B-A)}$
4. ${\displaystyle B_{2}\left(a+{\frac {7}{8}}(b-a)\right)=B+{\frac {1}{8}}(B-A)}$
5. ${\displaystyle B_{2}(b)=B}$

${\displaystyle B_{3}}$is defined as a linear function by each piece of linear transformed so as to to have transformed by new values of , , and employs so and correspond to the corner points of the linear piece. ${\displaystyle B_{2}}$ ${\displaystyle B_{1}}$ ${\displaystyle B_{2}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle (a,A)}$ ${\displaystyle (b,B)}$

${\displaystyle B_{k}}$we define for any by each piece of linear transformed so as to to have transformed by new values of , , and employs so and correspond to the corner points of the linear piece. ${\displaystyle k\in \mathbb {N} ,k\geq 2}$ ${\displaystyle B_{k-1}}$ ${\displaystyle B_{1}}$ ${\displaystyle B_{2}}$ ${\displaystyle a}$ ${\displaystyle b}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle (a,A)}$ ${\displaystyle (b,B)}$

The Bolzanofunktion is the pointwise limit of the sequence of functions: . ${\displaystyle B}$ ${\displaystyle B(x)=\lim _{k\to \infty }B_{k}(x)}$

source

Johan Thim: Continuous Nowhere Differentiable Functions. (pdf; 650 kB) Master's thesis, Luleå University of Technology. October 2003, pp. 11-17 , accessed on September 16, 2013 (English).