Remarks on solutions to a generalization of the radical functional equations (Q1784269)

During the \(16\)th ``International Conference on Functional Equations and Inequalities'' a talk was given concerning the stability of the so-called radical functional equation, that is, \[ f (\sqrt{x^{2} + y^{2}}) = f (x) + f (y). \] The second author's question about the general solution of the equation itself was answered later by the first one. Contrary to some assertions in the literature the general solution is not an arbitrary quadratic function, but rather an arbitrary function of the form \[ x\longmapsto a(x^{2}), \] with an appropriate additive function \(a\). This paper contains interesting generalizations and applications of the above result.