Український математичний конгрес  2009
Александр Харазишвили (Тбилисский Математический Институт им. А. Размадзе, Тбилиси, Грузия) On continuous totally nonmonotone 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, NorthHolland, Amsterdam, 1977), for every function f acting from A into B, there exists an infinite subset X of A such that the restriction fX 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 nonmonotone if there exists no subset X of A with card(X) = card(A) such that gX is monotone. For instance, under the Continuum Hypothesis, every SierpinskiZygmund function (see K. Kuratowski, Topology, vol. 1, Academic Press, New York, 1966) is totally nonmonotone. 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 nonmonotone 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 sigmaideal of subsets of A) are underlined.
