caa a22 4500 465825079 CHVBK 20180323112147.0 cr unu---uuuuu 170327e19900401xx s 000 0 eng 10.1007/BF01833143 doi (NATIONALLICENCE)springer-10.1007/BF01833143 Baran M. Department of Mathematics, Pedagogical University of Cracow, Podchorazych, 30-084, Cracow, Poland aut On the graph of a quasi-additive function [Elektronische Daten] [M. Baran] Summary: LetX and (Y, |·|) denote a linear topological space (real or complex) and a normed linear space (real or complex), respectively. A mapf: X → Y is said to be quasi-additive if|f(x+y) − f(x) − f(y)| ⩽ ε min{|f(x+y)|, |f(x) + f(y)|} (1) forx, y ∈X with the constant 0 ⩽ε < 1. The class of functions which satisfy (1) was introduced by Tabor in [3] (forX=Y=ℝ, see also [5] p. 121) and [4] (for normed spaces). It is obvious that every additive map is quasi-additive (withε = 0). Tabor proved that quasi-additive functions have properties similar to additive functions. In this paper we consider the case whereY=ℝ. The main result of this paper is as follows. Theorem 1.Let X be a locally convex space. If f: X → ℝis a discontinuous quasi-additive function then the set Gr(f) is dense in X × ℝ(in the projective topology). Here Gr(f) := {(x, t) ∈X × ℝ: x ∈ X, t = f(x)}. This result extends the known property of real additive functions (see [2], pp. 276-277) to the class of real quasi-additive functions. This gives a positive answer (in caseX = ℝ) to a question posed by Tabor [3]. As an application, by de Bruijn's theorem (see [1]), we get Theorem 2.Let f: ℝ → ℝbe a quasi-additive function. The following conditions are equivalent: (a) f is continuous or f is additive; (b) for every y ∈ ℝ the function ℝ ∋ x → f(x + y) − f(x) is continuous. Birkhäuser Verlag, 1990 Primary 39C05 nationallicence Secondary 9A11 nationallicence aequationes mathematicae Birkhäuser-Verlag 39/2-3(1990-04-01), 129-133 0001-9054 39:2-3<129 1990 39 10 https://doi.org/10.1007/BF01833143 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/BF01833143 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Baran M. Department of Mathematics, Pedagogical University of Cracow, Podchorazych, 30-084, Cracow, Poland aut NATIONALLICENCE 773 0- aequationes mathematicae Birkhäuser-Verlag 39/2-3(1990-04-01), 129-133 0001-9054 39:2-3<129 1990 39 10 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer