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
80 Gyula Maksa 1 U+V U V - I A uA vf + J fo- J fo- I fo 0 0 0 0 u-f-1 u-\-t-\-v u-\-t u+v u = J A tA vf- J f 0+ J f 0+ J fo~ J fou 0 0 0 0 Since A tA vf and fo are continuous functions, the right hand side is continuously differentiable with respect to u then so is the left hand side. Differentiating both sides with respect to u and taking into consideration (4) we obtain that Q -r-[B(u + t,v) - = A tA vAif(u) - f 0{u + t + v) + fo{u + t) ou + fo(u + v)~ fo(u) = A tA v(fo + a)(u) - A tA vfo(u) = A tA va(u) = 0 (a being additive). Therefore B(u + t,v)~ B(u , v) = B(t, v) - B( 0, v) = B(t , v), that is, i? is additive in its first (and by the symmetry also in its second) variable. Thus, it is well-known (see [2]) and easy to see that, the function ÍV: R —> R defined by N(u) = \B(u,u), ii E R is quadratic and (6) B{u,v) = N(u + v)-N(u)-N(v) u,v £ R. Define the function H:TL 3 —> R by (3) and apply Lemma 2 to get the continuity of the function (x,u) — > H(x,u,v), (x,u) G R 2 for all fixed v G R. This implies that the function s: R 2 —> R defined by l «(«.») = J H(x,u,v)dx (ii, v) 6 R 2 0 is continuous in its first variable (for all fixed v G R). Therefore, by (3), (5) and (6) we have l s(u, v) = J A uA vf - f(u + v) + f(u) + f(v) 0
