# zbMATH — the first resource for mathematics

The continuity of additive and convex functions which are upper bounded on non-flat continua in $$\mathbb R^n$$. (English) Zbl 1428.26022
Summary: We prove that for a continuum $$K\subset \mathbb R^n$$ the sum $$K^{+n}$$ of $$n$$ copies of $$K$$ has non-empty interior in $$\mathbb R^n$$ if and only if $$K$$ is not flat in the sense that the affine hull of $$K$$ coincides with $$\mathbb R^n$$. Moreover, if $$K$$ is locally connected and each non-empty open subset of $$K$$ is not flat, then for any (analytic) non-meager subset $$A\subset K$$ the sum $$A^{+n}$$ of $$n$$ copies of $$A$$ is not meager in $$\mathbb R^n$$ (and then the sum $$A^{+2n}$$ of $$2n$$ copies of the analytic set $$A$$ has non-empty interior in $$\mathbb R^n$$ and the set $$(A-A)^{+n}$$ is a neighbourhood of zero in $$\mathbb R^n)$$. This implies that a mid-convex function $$f\colon D\to\mathbb R$$ defined on an open convex subset $$D\subset\mathbb R^n$$ is continuous if it is upper bounded on some non-flat continuum in $$D$$ or on a non-meager analytic subset of a locally connected nowhere flat subset of $$D$$.

##### MSC:
 26B05 Continuity and differentiation questions 54D05 Connected and locally connected spaces (general aspects) 26B25 Convexity of real functions of several variables, generalizations 54C05 Continuous maps 54C30 Real-valued functions in general topology
Full Text:
##### References:
 [1] T. Banakh and E. Jabłońska, Null-finite sets in metric groups and their applications, Israel J. Math. 230 (2019), 361-386. [2] F. Bernstein, G. Doetsch, Zur Theorie der konvexen Funktionen, Math. Ann. 76 (1915), 514-526. · JFM 45.0627.02 [3] R. Engelking, Theory of Dimensions, Finite and Infinite, Sigma Series in Pure Mathematics, vol. 10, Lemgo, Heldermann Verlag, 1995. · Zbl 0872.54002 [4] P. Erdős, On some properties of Hamel bases, Colloq. Math. 10 (1963), 267-269. [5] R. Ger, Some remarks on convex functions, Fund. Math. 66 (1969), 255-262. · Zbl 0192.41101 [6] R. Ger, Thin sets and convex functions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 413-416. · Zbl 0268.26009 [7] R. Ger and Z. Kominek, Boundedness and continuity of additive and convex functionals, Aequationes Math. 37 (1989), 252-258. · Zbl 0702.46004 [8] R. Ger and M. Kuczma, On the boundedness and continuity of convex functions and additive functions, Aequationes Math. 4 (1970), 157-162. · Zbl 0194.17402 [9] W. Holsztyński, Universality of mappings onto the products of snake-like spaces. Relation with dimension, Bull. Acad. Polon. Sci. Sr. Sci. Math. Astronom. Phys. 16 (1968), 161-167. [10] W. Jabłoński, Steinhaus-type property for a boundary of a convex body, J. Math. Anal. Appl. 447 (2019), 769-775. [11] R.R. Kallman and F.W. Simmons, A theorem on planar continua and an application to authomorphisms of the field of complex numbers, Topology Appl. 20 (1985), 251-255. · Zbl 0572.12007 [12] A.S. Kechris, Classical Descriptive Set Theory, Springer, New York, 1998. · Zbl 0819.04002 [13] Z. Kominek, On the sum and difference of two sets in topological vector spaces, Fund. Math. 71 (1971), 165-169. · Zbl 0214.37403 [14] M. Kuczma, An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, second edition (A. Gilànyi, ed.), Birkhäuser Verlag, Basel, 2009. · Zbl 1221.39041 [15] M.E. Kuczma, On discontinuous additive functions, Fund. Math. 66 (1969/1970), 383-392. · Zbl 0194.36003 [16] K. Kuratowski, Topology I, Academic Press, 1966. [17] S. Kurepa, Convex functions, Glasnik Mat.-Fiz. Astronom. 11 (1956), no. 2, 89-93. · Zbl 0072.05301 [18] I.K. Lifanov, The dimension of a product of one-dimensional bicompacta, Dokl. Akad. Nauk SSSR 180 (1968), 534-537 (in Russian); English transl.: Soviet Math. Dokl. 9 (1968), 648-651. [19] M.R. Mehdi, On convex functions, J. London Math. Soc. 39 (1964), 321-326. · Zbl 0125.06304 [20] A. Ostrowski, Über die Funkionalgleichung der Exponentialfunktion und verwandte Funktionalgleichungen, Jber. Deutsch. Math.-Verein 38 (1929), 54-62. · JFM 55.0800.01 [21] B.J. Pettis, Remarks on a theorem of E.J. McShane, Proc. Amer. Math. Soc. 2 (1951), 166-171. · Zbl 0043.05502 [22] S. Piccard, Sur les ensembles de distances des ensembles de points d’un espace Euclidien, Mémoires de l’Universté Neuchâtel, vol. 13, Secrétariat Univ., Neuchâtel, 1939. · JFM 65.1170.03 [23] Ja. Tabor, Jo. Tabor and M. Żołdak, Approximately convex functions on topological vector spaces, Publ. Math. Debrecen 77 (2010), 115-123.
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.