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Cantor function

Cantor function

Cantor function was first discovered by Henry John Stephen Smith and invented by Georg Cantor in 1883. Cantor set is a set of point that lies on single line segment. Cantor is one of the mathematician who laid foundation for the modern point set topology.

Cantor set is obtained from the closed interval 0 to 1. Remove the middle third from the interval, that is 1/3, 2/3. Remove the remaining middle third from the remaining sets that is 1/9, 2/9, continue the process infinitely. Left over points forms the fractal called the Standard Cantor Set.

As cantor set is the set of points not excluded, the length of the original interval is removed in each step. In the first step, the length of interval removed is 1/3. In the second interval, length removed is 1/9. At fourth step, four intervals is removed with total length of 1/27. So the total length removed in the geometric series is

SUM1 to infinity (2n-1/3n) = 1

The proportion left is 1 – 1 = 0.

Cantor set has left over with Zero interval and the set is uncountable. Sum of the length of the removed interval is equal to the original interval. The process of removing the middle third's is an example of finite subdivision rule.

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