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Ramsey type theorems for real functions
Author(s) -
Buczolich Z.
Publication year - 1989
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s0025579300013632
Subject(s) - mathematics , uncountable set , concave function , combinatorics , countable set , regular polygon , bounded function , discrete mathematics , set function , graph , function (biology) , type (biology) , element (criminal law) , interval (graph theory) , ramsey's theorem , set (abstract data type) , mathematical analysis , geometry , evolutionary biology , computer science , political science , law , biology , programming language , ecology
Ramsey's theorem implies that every function f :0, 1 ℝ isconvex or concave on an infinite set. We show that there is an upper semicontinuous function which is not convex or concave on any uncountable set. We investigate those functions which are not convex on any r element set (r). A typical result: if f is bounded from below and is not convex on any infiniteset then there exists an interval on which the graph of f can be covered by the graphs of countably many strictly concave functions.