The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathologicalcontinuous
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 .
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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. Japan1, 176-177, 1903.Tall, D. O. "The
Blancmange Function, Continuous Everywhere but Differentiable Nowhere." Math.
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Math. Teaching111, 48-52, 1985.Wells, D. The
Penguin Dictionary of Curious and Interesting Geometry. London: Penguin,
pp. 16-17, 1991.
th iteration contains 
points, where 
, and can be obtained by setting 
, letting
![b(m+2^(n-1))=2^n+1/2[b(m)+b(m+2^n)],](/images/equations/BlancmangeFunction/NumberedEquation1.svg)
to 1 by steps of 
and 
to 
by steps of 
.
