Editing Cantor function (section)

==Properties==
The Cantor function challenges naive intuitions about [[continuous function|continuity]] and [[measure (mathematics)|measure]]; though it is continuous everywhere and has zero derivative [[almost everywhere]], <math display="inline">c(x)</math> goes from 0 to 1 as <math display="inline>x</math> goes from 0 to 1, and takes on every value in between.  The Cantor function is the most frequently cited example of a real function that is [[uniformly continuous]] (precisely, it is [[Hölder continuous]] of  exponent <math>\alpha = \log_3(2)</math>) but not [[absolute continuity|absolutely continuous]]. It is constant on intervals of the form (0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>022222..., 0.''x''<sub>1</sub>''x''<sub>2</sub>''x''<sub>3</sub>...''x''<sub>n</sub>200000...), and every point not in the Cantor set is in one of these intervals, so its derivative is 0 outside of the Cantor set. On the other hand, it has no [[derivative]] at any point in an [[uncountable]] subset of the [[Cantor set]] containing the interval endpoints described above.

The Cantor function can also be seen as the [[cumulative distribution function|cumulative probability distribution function]] of the 1/2-1/2 [[Bernoulli measure]] ''μ'' supported on the Cantor set: <math display="inline">c(x)=\mu([0,x])</math>. This probability distribution, called the [[Cantor distribution]], has no discrete part. That is, the corresponding measure is [[Atom (measure theory)|atomless]]. This is why there are no jump discontinuities in the function; any such jump would correspond to an atom in the measure.

However, no non-constant part of the Cantor function can be represented as an integral of a [[probability density function]]; integrating any putative [[probability density function]] that is not [[almost everywhere]] zero over any interval will give positive probability to some interval to which this distribution assigns probability zero. In particular, as {{harvtxt|Vitali|1905}} pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere.

The Cantor function is the standard example of a [[singular function]].

The Cantor function is also a standard example of a function with [[bounded variation]] but, as mentioned above, is not absolutely continuous. However, every absolutely continuous function is continuous with bounded variation. 

The Cantor function is non-decreasing, and so in particular its graph defines a [[rectifiable curve]].  {{harvtxt|Scheeffer|1884}} showed that the arc length of its graph is 2. Note that the graph of any nondecreasing function such that <math>f(0)=0</math> and <math>f(1)=1</math> has length not greater than 2. In this sense, the Cantor function is extremal.

===Lack of absolute continuity===
The [[Lebesgue measure]] of the [[Cantor set]] is 0. Therefore, for any positive ''ε''&nbsp;<&nbsp;1 and any ''δ'' > 0, there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length <&nbsp;''δ'' over which the Cantor function cumulatively rises more than&nbsp;''ε''.

In fact, for every ''δ''&nbsp;>&nbsp;0 there are finitely many pairwise disjoint intervals (''x<SUB>k</SUB>'',''y<SUB>k</SUB>'') (1&nbsp;≤&nbsp;''k''&nbsp;≤&nbsp;''M'') with <math>\sum\limits_{k=1}^M (y_k-x_k)<\delta</math> and <math>\sum\limits_{k=1}^M (c(y_k)-c(x_k))=1</math>.