Rudin‚s problem on groups and a generalization of mean value theorem

Summary.

We consider two functional equations: one (11) $f(x) - f(y) = h(sx + ty)(x - y)$, for $x,y \in \mathbb{R},\, x \neq y$ connected to Rudin‚s problem on groups and the other (12) $f(x) - g(y) = (x - y)(h(x + y) + k(x) + \ell(y))$, for $x,y \in \mathbb{R}$, a generalization of the mean value theorem.