Cantor singular function

Another mathematical monster , sometimes called the Cantor singular function [11], is closely related to the Cantor set. This singular function is constructed by integrating an appropriate distribution function defined on the Cantor set. One such distribution function is constructed by first considering the uniform distribution of mass on the interval [0,1], with total mass equal to one (in some arbitrary units). For example, one could visualize the initiator /q of Figure 2.1, not as a line segment, but as a bar of some material with unit mass density po and length /o = 1. The operation... 

Figure 2J2 Intermediate stage (n = 6) in the construction of the Cantor singular function or Devil s staircase (top) and schematic illustration of the self-affinity of the Cantor singular function or DevU s staircase (bottom) the enlargement is identical to the original, but the enlargement (scaling) factors are different in the x and y directions.

The above definition immediately proved unsatisfactory, in that it excluded a number of sets with properties very similar to those of sets that satisfied the definition and, therefore, which also ought to be regarded as fractals. Indeed, according to this definition, the Cantor singular function (Dh = A- = 1) and the Peano planefilling curve (Dh = A = 2) are not fractals. Various other definitions of fractals have been proposed [e.g. 4 (p. 362)], but they all seem to suffer from the same drawback. 

Close inspection of Figure 2.7c reveals that c(t) is self-affine, like Cantor s singular function [10 (p. 29)]. Both w(t) and c(t) are functions of a single variable. 

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