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==Definition==
[[File:Cantor function.gif|300pxthumb|rightupright=1.35|Iterated construction of the Cantor function]]
See figure. To formally define the Cantor function ''<math>c '' : [0,1] → \to[0,1] </math>, let ''<math>x ''</math> be any number in <math>[0,1] </math> and obtain ''<math>c ''( ''x '') </math> by the following steps: ▼
#Express ''<math>x ''</math> in base 3. ▼▲See figure. To formally define the Cantor function ''c'' : [0,1] → [0,1], let ''x'' be in [0,1] and obtain ''c''(''x'') by the following steps:
#If ''the base-3 representation of <math>x ''</math> contains a 1, replace every digit strictly after the first 1 by 0. ▼
▲#Express ''x'' in base 3.
▲#If ''x'' contains a 1, replace every digit strictly after the first 1 by 0.
#Replace any remaining 2s with 1s.
#Interpret the result as a binary number. The result is ''<math>c''(''x'')</math>.
For example:
* 1<math>\tfrac14</4math> ishas the ternary representation 0.02020202... in base 3. There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... WhenThis readis inthe basebinary 2,representation thisof corresponds to 1<math>\tfrac13</3math>, so ''<math>c''(1/4\tfrac14) = 1\tfrac13</3math>.
* 1<math>\tfrac15</5math> ishas the ternary representation 0.01210121... in base 3. The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since thereit arehas no 2s. WhenThis readis inthe basebinary 2,representation thisof corresponds to 1<math>\tfrac14</4math>, so ''<math>c''(1/5\tfrac15) = 1\tfrac14</4math>.
* <math>\tfrac{200/}{243}</math> ishas the ternary representation 0.21102 (or 0.211012222...) in base 3. The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. WhenThis readis inthe basebinary 2,representation this corresponds toof 3<math>\tfrac34</4math>, so ''<math>c''(\tfrac{200/}{243}) = 3\tfrac34</4math>.
Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function ''c''<nowikimath> c: [0,1] → \to[0,1]</math> can be defined as</nowiki>
:<math>c(x) =\begin{cases}
</math>
This formula is well-defined, since every member of the Cantor set has a ''unique'' base 3 representation that only contains the digits 0 or 2. (For some members of <math>\mathcal{C}</math>, the ternary expansion is repeating with trailing 2's and there is an alternative non-repeating expansion ending in 1. For example, 1<math>\tfrac13</3math> = 0.1<sub>3</sub> = 0.02222...<sub>3</sub> is a member of the Cantor set). Since ''<math>c''(0) = 0</math> and ''<math>c''(1) = 1</math>, and ''<math>c''</math> is monotonic on <math>\mathcal{C}</math>, it is clear that <math>0\le ≤ ''c''(''x'') ≤\le 1</math> also holds for all <math>x\in[0,1]\setminus\mathcal{C}</math>.
==Properties==
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