Editing Cantor function (section)

=== Fractal volume ===
The Cantor function is closely related to the [[Cantor set]]. The Cantor set ''C'' can be defined as the set of those numbers in the interval [0,&nbsp;1] that do not contain the digit 1 in their [[radix|base]]-3 (triadic) expansion, except if the 1 is followed by zeros only (in which case the tail 1000<math>\ldots</math> can be replaced by 0222<math>\ldots</math> to get rid of any 1). It turns out that the Cantor set is a [[fractal]] with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume).  Only the ''D''-dimensional volume <math> H_D </math> (in the sense of a [[Hausdorff dimension|Hausdorff-measure]]) takes a finite value, where <math> D = \log_3(2) </math> is the fractal dimension of ''C''.  We may define the Cantor function alternatively as the ''D''-dimensional volume of sections of the Cantor set

: <math>
f(x)=H_D(C \cap (0,x)). 
</math>