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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 ''α'' = log 2/log 3) 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 scheme|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.
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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.
===Lack of absolute continuity===
Because the [[Lebesgue measure]] of the [[Uncountable set|uncountably infinite]] [[Cantor set]] is 0, for any positive ''ε'' and ''δ'', there exists a finite sequence of [[pairwise disjoint]] sub-intervals with total length < ''δ'' over which the Cantor function cumulatively rises more than ''ε''. If a given sequence did not, all but an arbitrarily short portion of each sub-interval containing a point in the Cantor set could be discarded and a finite number equal-length sub-intervals containing different points in the Cantor set added, iteratively, until the cumulative rise was greater than ''ε''.
== Alternative definitions ==
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