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Blancmange Function (Tagaki Curve)

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The blancmange function (also called the Tagaki function or curve) is a bumpy, pudding-shaped, nowhere differentiable function.

blancmange function

Graph of the blancmange function. It’s named because it has the shape of a traditional English dessert.


Originally called the Tagaki function, after its creator, it was renamed the blancmange function by Tall (1982).

Notation for the Blancmange Function

The blancmange function is defined as the infinite series (Kearnes, n.d.):
definition tagaki function

Where h(x) = inf{|x−n| | n ∈ ℤ} is a sawtooth function of period 1.

In simpler terms, the blancmange function is like a fractal, made up of an infinite number of smaller “blancmanges.” It can be constructed by adding up sawtooth function iterations to infinity (Tall, 2013).

sawtooth to blancmange

The first two iterations of the sawtooth function, superimposed on the final blancmange. The sawtooth will get closer and closer to the shape, superimposing the blancmange completely after an infinite number of iterations.


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.
Kearnes, K. The blancmange function. https://math.colorado.edu/~kearnes/Teaching/Courses/F18/Blancmange.pdfeitgen, 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.
Tall, D. The Blancmange F. Continuous Everywhere but Differentiable Nowhere. The Mathematical Gazette. 66. 10.2307/3617301.
Tall, D. How Humans Learn to Think Mathematically: Exploring the Three Worlds of Mathematics. Cambridge University Press, 2013.
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 16-17, 1991.


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Stephanie Glen. "Blancmange Function (Tagaki Curve)" From CalculusHowTo.com: Calculus for the rest of us! https://www.calculushowto.com/blancmange-function-tagaki-curve/
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