Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1998. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 25)

MAKSA, GY ., Functions having quadratic differences in a given class

Functions having quadratic differences in a given class GYULA MAKSA Abstract. Starting from a problem of Z. Daróczy we define the quadratic difference property and show that the class of all real-valued continuous functions on R and some of its subclasses have this property while the class of all bounded functions does not have. 1. Introduction For a function /:R —• R (the reals) and for a fixed y £ R define the function A

yf on R by A

yf(x) = f(x + y) — f{x), x £ R. The functions A. N

: R —> R are said to be additive and quadratic if A(x + y) = A(x) + A(y) x, y £ R and N(x + y) + N{x-y) = 2N{x) + 2N(y) x, y £ R, respectively. It is well-known (see [1], [5], [2]) that, if an additive function is bounded from one side on an interval of positive length then A(x) = cx

, x £ R for some c £ R and there axe discontinuous additive functions. Similarly, if a quadratic function is bounded on an interval of positive length then N(x) = dx

2

, x £ R for some d £ R and there are discontinuous quadratic functions. In [4] Z. DAR

Ó

CZY asked that for which properties T the following statement is true: (*) Let /:R —>• R be a function such that for all fixed y £ R the function AyA-yf has the property T. Then (1) f=f* + N + A on R This research has been supported by grants from the Hungarian National Foundation for Scientific Research (OT'KA) (No. T-016846) and from the Hungarian High Educational Research and Development Fund (FKFP) (No. 0310/1997).