Cantor function

It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure.

Though it is continuous everywhere and has zero derivative almost everywhere, its value still goes from 0 to 1 as its argument reaches from 0 to 1.

[4] Georg Cantor (1884) introduced the Cantor function and mentioned that Scheeffer pointed out that it was a counterexample to an extension of the fundamental theorem of calculus claimed by Harnack.

The Cantor function was discussed and popularized by Scheeffer (1884), Lebesgue (1904) and Vitali (1905).

can be defined as This formula is well-defined, since every member of the Cantor set has a unique base 3 representation that only contains the digits 0 or 2.

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 probability distribution function of the 1/2-1/2 Bernoulli measure μ supported on the Cantor set:

The Cantor function is non-decreasing, and so in particular its graph defines a rectifiable curve.

Because the Lebesgue measure of the uncountably infinite Cantor set is 0, for any positive ε < 1 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.

In fact, for every δ > 0 there are finitely many pairwise disjoint intervals (xk,yk) (1 ≤ k ≤ M) with

One may check that fn converges pointwise to the Cantor function defined above.

Indeed, separating into three cases, according to the definition of fn+1, one sees that If f denotes the limit function, it follows that, for every n ≥ 0, The Cantor function is closely related to the Cantor set.

It turns out that the Cantor set is a fractal with (uncountably) infinitely many points (zero-dimensional volume), but zero length (one-dimensional volume).

Define the reflection as The first self-symmetry can be expressed as where the symbol

For the left and right magnifications, write the left-mappings Then the Cantor function obeys Similarly, define the right mappings as Then, likewise, The two sides can be mirrored one onto the other, in that and likewise, These operations can be stacked arbitrarily.

Adding the subscripts C and D, and, for clarity, dropping the composition operator

Thus, every dyadic rational is in one-to-one correspondence with some self-symmetry of the Cantor function.

Some notational rearrangements can make the above slightly easier to express.

stand for L and R. Function composition extends this to a monoid, in that one can write

It can be viewed as a finite number of left-right moves down an infinite binary tree; the infinitely distant "leaves" on the tree correspond to the points on the Cantor set, and so, the monoid also represents the self-symmetries of the Cantor set.

In fact, a large class of commonly occurring fractals are described by the dyadic monoid; additional examples can be found in the article on de Rham curves.

Other fractals possessing self-similarity are described with other kinds of monoids.

In particular, it obeys the exact same symmetry relations, although in an altered form.

This expansion is discussed in greater detail in the article on the dyadic transformation.

In general, for any z < 1/2, Cz(y) looks like the Cantor function turned on its side, with the width of the steps getting wider as z approaches zero.

This line of research was started in the 1990s by Darst,[5] who showed that the Hausdorff dimension of the set of non-differentiability of the Cantor function is the square of the dimension of the Cantor set,

Subsequently Falconer[6] showed that this squaring relationship holds for all Ahlfors's regular, singular measures, i.e.

Later, Troscheit[7] obtain a more comprehensive picture of the set where the derivative does not exist for more general normalized Gibb's measures supported on self-conformal and self-similar sets.

Hermann Minkowski's question mark function loosely resembles the Cantor function visually, appearing as a "smoothed out" form of the latter; it can be constructed by passing from a continued fraction expansion to a binary expansion, just as the Cantor function can be constructed by passing from a ternary expansion to a binary expansion.

The question mark function has the interesting property of having vanishing derivatives at all rational numbers.