Blancmange Function

A Continuous Function which is nowhere Differentiable. The iterations towards the continuous function are Batrachions resembling the Hofstadter-Conway $10,000 Sequence. The first six iterations are illustrated below. The $d$th iteration contains $N+1$ points, where $N=2^d$, and can be obtained by setting $b(0)=b(N)=0$, letting

$$b(m+2^{n-1})=2^n+{\textstyle{1\over 2}}[b(m)+b(m+2^n)],$$

and looping over $n=d$ to 1 by steps of $-1$ and $m = 0$ to $N-1$ by steps of $2^n$.

Peitgen and Saupe (1988) refer to this curve as the Takagi Fractal Curve.

See also Hofstadter-Conway $10,000 Sequence, Weierstraß Function

References

Dixon, R. Mathographics. New York: Dover, pp. 175-176 and 210, 1991.

Peitgen, H.-O. and Saupe, D. (Eds.). ``Midpoint Displacement and Systematic Fractals: The Takagi Fractal Curve, Its Kin, and the Related Systems.'' §A.1.2 in The Science of Fractal Images. New York: Springer-Verlag, pp. 246-248, 1988.

Takagi, T. ``A Simple Example of the Continuous Function without Derivative.'' Proc. Phys. Math. Japan 1, 176-177, 1903.

Tall, D. O. ``The Blancmange Function, Continuous Everywhere but Differentiable Nowhere.'' Math. Gaz. 66, 11-22, 1982.

Tall, D. ``The Gradient of a Graph.'' Math. Teaching 111, 48-52, 1985.