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
78 Gyula Maksa where /* has the property T, TV is a quadratic function and A is an additive function. In this note we prove that, if T is the fc-times continuously differentiability (k > 0 integer) or Ar-times differentiability (k > 0 integer or k = + oo) or being polynomial then the statement (*) is true while if T is the boundedness then (*) is not true. The following lemma will play an important role in our investigations. Lemma 1. For all functions /: R R and for all u, v,x £ R we have The proof is a simple computation therefore it is omitted. An other basic tool we will use is the following result of DE BRUIJN ([3] Theorem 1.1.) Theorem 1. Suppose that f: R —> R is a function such that the function Ay/ is continuous for all fixed y £ R. Then f — f* + A on R with some continuous /*: R —R and additive A \ R —> R. Finally we will need the following two lemmata. Lemma 2. Let /: R —> R be a function such that A uA vf is continuous for all fixed u, v E R. Define (3) H(x, u, v) = A uA vf(x) - f{u + v) + f(u) + f(v) x, u, v £ R. Then the function (x,u) —• ff(x,u, v), (x :u) £ R 2 is continuous for all fixed Proof. Let v £ R be fixed. Since A u(A vf) is continuous for all fixed u £ R, Theorem 1 implies that A vf = f* + A v on R where /*: R —> R is continuous and A v is additive. Thus, by (3), 2. Preliminary results (2) A uA vf(x) = v £ R. H(x, u, v) = A vf(x -f u) - A vf(x) - A vf(u) + f(v) = /:('•r + u) - r v(x) - f*(u) + f(v) whence the continuity of (x,u) —> H(x :u,v), (x,u) £ R 2 follows.
