Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-01T07:04:49.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Ramsey type theorems for real functions

Part of: Real functions

Published online by Cambridge University Press:  26 February 2010

Z. Buczolich
Z. Buczolich
Affiliation:
Department of Analysis, Etvs Lornd University, Budapest, Mǔzeum kit. 6-8, H-1088, Hungary.
Get access

Abstract

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.

MSC classification

Secondary: 26A51: Convexity, generalizations
Type
Research Article
Information
Mathematika , Volume 36 , Issue 1 , June 1989 , pp. 131 - 141
Copyright
Copyright University College London 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

ABLP.Agronsky, S.Bruckner, A. M.Laczkovich, M. and Preiss, D.. Convexity conditionsand intersections with smooth functions. Trans. Amer. Math. Soc., 289 (1985).CrossRefGoogle Scholar
Bl.Blass, A.. A partition theorem for perfect sets. Proc. Amer. Math. Soc., 82 (1981).CrossRefGoogle Scholar
Bu.Buczolich, Z.. Sets of convexity of continuous functions. Ada Math. Hung., 53 (1988), 291303.CrossRefGoogle Scholar
EHMR.Erds, P., Hajnal, A., Mt, A. and Rad, R.. Combinatorial Set Theory: Partition Relations for Cardinals (Akadmiai Kiad Budapest North Holland, 1984).Google Scholar
F.Filipczak, F. M.. Sur les fonctions continues relativement monotones. Fund. Math., 58 (1966), 7587.CrossRefGoogle Scholar
HS.Hewitt, E.. and Stromberg, K.. Real and Abstract Analysis (Springer, 1969).Google Scholar
S.Sierpiiisi, W.. Sur une proprit des ensembles F linaires. Fund. Math., 14 (1929), 216220.Google Scholar