Үзіліссіз, бірақ абсолютті үзділіссіз емес функция: Кантор функциясы

Continuous function that is not absolutely continuous

In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. 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. Thus, in one sense the function seems very much like a constant one which cannot grow, and in another, it does indeed monotonically grow. It is also called the Cantor ternary function, the Lebesgue function, Lebesgue's singular function, the Cantor–Vitali function, the Devil's staircase, the Cantor staircase function, and the Cantor–Lebesgue function. 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 , and .

The Cantor function challenges naive intuitions about continuity and measure; though it is continuous everywhere and has zero derivative almost everywhere, goes from 0 to 1 as 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 absolutely continuous. It is constant on intervals of the form (0. x1x2x3 xn022222 , 0. x1x2x3 xn200000 ), 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 probability distribution function of the 1/2 1/2 Bernoulli measure μ supported on the Cantor set: This probability distribution, called the Cantor distribution, has no discrete part. That is, the corresponding measure is 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 pointed out, the function is not the integral of its derivative even though the derivative exists almost everywhere. 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. showed that the arc length of its graph is 2. Note that the graph of any nondecreasing function such that and has length not greater than 2. In this sense, the Cantor function is extremal.

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 and .

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, 1] that do not contain the digit 1 in their base 3 (triadic) expansion, except if the 1 is followed by zeros only (in which case the tail 1000 can be replaced by 0222 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 (in the sense of a Hausdorff measure) takes a finite value, where is the fractal dimension of C. We may define the Cantor function alternatively as the D dimensional volume of sections of the Cantor set