Український математичний конгрес - 2009

Александр Харазишвили (Тбилисский Математический Институт им. А. Размадзе, Тбилиси, Грузия)

On continuous totally non-monotone functions

Let A and B be any two linearly ordered sets. If A is infinite, then according to Ramsey's combinatorial theorem (see, e.g., Handbook of Mathematical Logic, North-Holland, Amsterdam, 1977), for every function f acting from A into B, there exists an infinite subset X of A such that the restriction f|X is monotone.

In some special cases this result can be significantly strengthened. For example, in the case A = B = R, where R denotes the real line, every Lebesgue measurable (or possessing the Baire property) function f acting from R into R is monotone on some nonempty perfect subset of R.

A function g acting from A into B is called totally non-monotone if there exists no subset X of A with card(X) = card(A) such that g|X is monotone.

For instance, under the Continuum Hypothesis, every Sierpinski-Zygmund function (see K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966) is totally non-monotone. However, such a function is totally discontinuous in the sense that its restriction to any uncountable subset of R is not continuous.

In this connection, some examples of continuous totally non-monotone functions defined on uncountable subsets of a linearly ordered set A are presented and compared to each other. Close connections of such functions with Luzin type constructions of singular sets (with respect to a given sigma-ideal of subsets of A) are underlined.