Blancmange Function
The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous
function which is nowhere differentiable.
Its name derives from the resemblance of its first iteration to the shape of the
dessert commonly made with milk or cream and sugar thickened with gelatin.
The iterations towards the continuous function are batrachions resembling the Hofstadter-Conway
$10,000 sequence. The first six iterations are illustrated below. The 
th iteration contains
points, where 
, and can be
obtained by setting 
, letting
and looping over 
to 1 by steps of 
and 
to 
by steps of
.
SEE ALSO: Batrachion,
Hofstadter-Conway $10,000 Sequence,
Monsters of Real Analysis,
Nowhere Differentiable Function,
Pathological,
Stolarsky-Harborth
Constant,
Weierstrass Function
REFERENCES:
Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. Experimental
Mathematics in Action. Wellesley, MA: A K Peters, pp. 111-113, 2007.
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.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 16-17, 1991.
Referenced on Wolfram|Alpha:
Blancmange Function
CITE THIS AS:
Weisstein, Eric W. "Blancmange Function."
From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BlancmangeFunction.html
![b(m+2^(n-1))=2^n+1/2[b(m)+b(m+2^n)],](/National_Library/im_/http://mathworld.wolfram.com/images/equations/BlancmangeFunction/NumberedEquation1.gif)
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