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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.
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