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Devil staircase

Devil Staircase

Devil staircase is also referred as Cantor function. It is named after Georg Cantor. It is a classic mountain bike route in the West Highland Way from Glencoe to Kinlochleven, in Scotland. Cantor function is an example of real function that is uniformly continuous but not absolutely continuous.

Cantor function can be defined as c: [0,1] --> [0,1]. Let x be in [0,1]. Here c goes from 0 to 1 as x goes from 0 to 1 and takes on every value that goes in between. c(x) can be obtained from the following steps,

  • Express x in base 3.
  • If x contains a 1, replace every digit after the first 1 by 0.
  • Replace all 2s with 1s.
  • Interpret the result as a binary number. The result is c(x).

Let us see this function with an example,

The value of c(x) = x - 1.

1. Value of 1/4 in base 3 can be written as 0.02020202... There is no 1, so the next stage remains 0.02020202. Replace all 2's with 1's. So, it can be rewritten as 0.01010101. When read in base value 2, c (1/4) = 1/3.

2. 1/5 becomes 0.01210121... in base 3; the digits after the first 1 are replaced by 0s to produce 0.01000000...; this is not rewritten since there are no 2s; when read in base 2, this is 1/4 so c(1/5) = 1/4.

Cantor function is the best example of Singular function. A real valued function f on the interval [a, b] is said to be singular function.

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Daily Maths Topic Today - devil staircase.

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