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 81 B(u,v) + j fo- j /0 - j /0 - f{u + v) + f(u) + f(v) 0 0 0 N(u + v)~ f(u + v) - (N(u

) - /(tz)) - (N(v) - f(v)) jh-jh-jh 0 0 0 A „(/ - N)(u) - (N(v) - f(v)) + j fo- j fo- j fo 0 0 0 This implies that A

v(f — N) is continuous for all fixed dER and Theorem 1 can be applied again to get the decomposition f — N — /* -f A on R with some continuous /*:R —

^ R and additive function A

, that is, (1) holds and the proof is complete. Theorem 3. Let L be as in Lemma 3. Then L has the quadratic difference property. Proof. If L is the class of all continuous functions then the statement is proved by Theorem 2. In the remaining cases, since all functions in L are continuous, Theorem 2 implies the decomposition (1) with continuous /*, quadratic N and additive A. We now prove that /* E L. For all y E R we get from (1) that Therefore A

yA_

yf * E L for all fixed y E R. Applying (2) in Lemma 1 we obtain that A

u(A

vf

*) E L for all fixed u,v E R- Obviously A

vf* is continuous thus, by Lemma 3, A

vf* E L. Since /* is continuous, Lemma 3 can be applied again to get /* E L. Remark. The set of all bounded functions /:R —• R does not have the quadratic difference property. Indeed, let Applying the Lagrangian mean value theorem with fixed u,v,x E R we have (7) AyA^yf = A

yA_

yr + 2N(y). f(x)-x ln(£

2 + 1) + 2arctgz - 2z, x E R. (8) A

uA

vf(x) = uA

vf\i) = uvf"(rj)