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 79 Lemma 3. Suppuse that L is one of the classes of the real-valued functions defined on R which are k-times continuously differentiable for some k > 0 integer or k-times differentiable for some 1 < k < +oc or polynomials. If f: R —> R is continuous and A

yf £ L for all fixed y £ R then f £ L. Proof. If L is the class of the continuous functions (k = 0) or of the polynomials, furthermore / is continuous and A

yf £ L for all fixed y £ R then, by Theorem 1 and by [3] page 203, respectively, / = /* + A for some /* £ L and additive function A. Therefore, by continuity of /, A(x) = cx

, x £ R with some c £ R whence / £ L follows. The remaining statement of Lemma 3 is just Lemma 3.1. in [3]. 3. The main results For the formulation of our main results let us begin with the following Definition. A subset E of the set of all functions /: R —> R is said to have the quadratic difference property if for all /:R —* R, with A

yA_

y/ £ E for all y £ R, the decomposition (1) holds true on R where /* £ E, N is a quadratic function and A is an additive function. First we prove the following Theorem 2. The class of all continuous functions /: R —» R has the quadratic difference property. Proof. By (2) in Lemma 1 we have that A

uA

vf is continuous for all fixed it, v £ R. In particular, A

u(Ai/) is continuous for all fixed u £ R. Applying Theorem 1 to A if we have (4) A

lf = f

0+a on R with some continuous /o

: R —

^ R and tion D on R

2 by Obviously, B is symmetric. Now we variable. For all u, t and i>, we have additive a: R —> R. Define the func(u,v) £ R

2. show that B is additive in its first u + f Ii v (5) B(u,v) = j A

uA

vf - J fo + J /o + j /o, 0 0 1 B(u + t, v) - B(u, v) = J A

u+

tA

vf o U-ft+U U+t V I /0 + / fo + J fo 0 0 0