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单词 CantorFunction
释义

Cantor function


The Cantor functionMathworldPlanetmath is a canonical example of a singular function. It is based on the Cantor setMathworldPlanetmath, and is usually defined as follows. Let x be a real number in [0,1] with the ternary expansion 0.a1a2a3, then let N be if no an=1 andotherwise let N be the smallest value such that an=1. Next let bn=12an for all n<N and let bN=1. We define the Cantor function (or the Cantor ternary function) as

f(x)=n=1Nbn2n.

This function can be easily checked to be continuousMathworldPlanetmathPlanetmath and monotonic on [0,1] and also f(x)=0 almost everywhere (it is constant on the complement of the Cantor set), with f(0)=0 and f(1)=1. Anotherinteresting fact about this function is that the arclength of the graph is 2, hence the calculus arclength formulaMathworldPlanetmath does not work in this case.


Figure 1: Graph of the cantor function using 20 iterations.

This function, and functions similar to it are frequently called the Devil’s staircase. Such functions sometimes occur naturally in various areas of mathematics and mathematical physics and are not just a pathologicalMathworldPlanetmath oddity.

References

  • 1 H. L. Royden. . Prentice-Hall, Englewood Cliffs, New Jersey, 1988
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更新时间:2025/5/4 19:29:45