Editing Cantor function (section)

==Definition==
[[File:Cantor function.gif|thumb|upright=1.35|Iterated construction of the Cantor function]]
To 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, using digits 0, 1, 2.  
#If the base-3 representation of <math>x</math> contains a 1, replace every digit strictly after the first 1 with 0.
#Replace any remaining 2s with 1s.
#Interpret the result as a binary number. The result is <math>c(x)</math>.

For example:
*<math>\tfrac14</math> has the ternary representation 0.02020202... There are no 1s so the next stage is still 0.02020202... This is rewritten as 0.01010101... This is the binary representation of <math>\tfrac13</math>, so <math>c(\tfrac14)=\tfrac13</math>.
*<math>\tfrac15</math> has the ternary representation 0.01210121... The digits after the first 1 are replaced by 0s to produce 0.01000000... This is not rewritten since it has no 2s. This is the binary representation of <math>\tfrac14</math>, so <math>c(\tfrac15)=\tfrac14</math>.
*<math>\tfrac{200}{243}</math> has the ternary representation 0.21102 (or 0.211012222...). The digits after the first 1 are replaced by 0s to produce 0.21. This is rewritten as 0.11. This is the binary representation of <math>\tfrac34</math>, so <math>c(\tfrac{200}{243})=\tfrac34</math>.

Equivalently, if <math>\mathcal{C}</math> is the [[Cantor set]] on [0,1], then the Cantor function <math>c:[0,1]\to[0,1]</math> can be defined as

<math display="block">c(x) =\begin{cases} 
\displaystyle \sum_{n=1}^\infty \frac{a_n}{2^n}, 
& \displaystyle  \text{if } x = \sum_{n=1}^\infty 
\frac{2a_n}{3^n}\in\mathcal{C}\ \text{for}\ a_n\in\{0,1\};
\\   
\displaystyle \sup_{y\leq x,\, y\in\mathcal{C}} c(y), 
& \displaystyle \text{if } x\in [0,1] \setminus \mathcal{C}. 
\end{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, <math>\tfrac13</math> = 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]\smallsetminus\mathcal{C}</math>.