# zbMATH — the first resource for mathematics

On the boundedness, Christensen measurability and continuity of $$t$$-Wright convex functions. (English) Zbl 1313.26017
The main goal of this paper is the investigation of regularity properties of $$t$$-Wright convex functions.
The main theorem of the first section is a Bernstein-Doetsch type one concerning $$t$$-Wright convex functions. It states that local boundedness at point of a $$t$$-Wright convex function entails its continuity at the same point. As consequences of this theorem, similar regularity theorems are stated, when the local boundedness condition is replaced for boundedness on a set is of second category, or boundedness on a set is of positive Lebesgue measure.
In the second section similar theorems are stated as in the first one using Christensen measurability.

##### MSC:
 26A51 Convexity of real functions in one variable, generalizations
##### Keywords:
convexity; Jensen convexity; Wright convexity
Full Text:
##### References:
 [1] Bernstein, F.; Doetsch, G., Zur theorie der konvexen funktionen, Math. Ann., 76, 514-526, (1915) · JFM 45.0627.02 [2] Brzdęk, J., The christensen measurable solutions of a generalization of the gołab-schinzel functional equation, Ann. Polon. Math., 64, 195-205, (1996) · Zbl 0860.39034 [3] Christensen, J. P. R., On sets of Haar measure zero in abelian Polish groups, Israel J. Math., 13, 255-260, (1972) · Zbl 0249.43002 [4] Fischer, P.; Słodkowski, Z., Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc., 79, 449-453, (1980) · Zbl 0444.46010 [5] Gajda, Z., Christensen measurability of polynomial functions and convex functions of higher orders, Ann. Polon. Math., XLVII, 25-40, (1986) · Zbl 0616.28006 [6] Ger, R., Convex functions of higher orders in Euclidean spaces, Ann. Polon. Math., 25, 293-302, (1972) · Zbl 0233.26010 [7] Ger, R., $$n$$-convex functions in linear spaces, Aequationes Math., 10, 172-176, (1974) · Zbl 0287.26013 [8] Ger, R.; Kuczma, M., On the boundedness and continuity of convex functions and additive functions, Aequationes Math., 4, 157-162, (1970) · Zbl 0194.17402 [9] Jabłońska, E., Jensen convex functions bounded above on nonzero christensen measurable sets, Annal. Math. Sil., 23, 53-55, (2009) · Zbl 1229.28027 [10] Kominek, Z., A continuity result on $$t$$-wright convex functions, Publ. Math. Debrecen, 63, 213-219, (2003) · Zbl 1053.26007 [11] Z. Kominek, Convex Functions in Linear Spaces, Prace Naukowe Uniwersytetu Śląskiego w Katowicach 1087 (Katowice, 1989). · Zbl 0699.46005 [12] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, Polish Scientific Publishers and Silesian University Press (Warszawa-Kraków-Katowice, 1985). · Zbl 0555.39004 [13] Maksa, Gy.; Nikodem, K.; Páles, Zs., Result on $$t$$-wright convexity, C.R. Math. Rep. Acad. Sci. Canada, 13, 274-278, (1991) · Zbl 0749.26007 [14] Olbryś, A., On the measurability and the Baire property of $$t$$-wright convex functions, Aequationes Math., 68, 28-37, (2004) · Zbl 1067.26009 [15] Olbryś, A., Some conditions implying the continuity of $$t$$-wright convex functions, Publ. Math. Debrecen, 68, 401-418, (2006) · Zbl 1109.26002 [16] Olbryś, A., Representation theorems for $$t$$-wright convexity, J. Math. Anal. Appl., 384, 273-283, (2011) · Zbl 1234.26036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.