On a symmetric functional equation in two variables (Q1884747)

The general differentiable solutions \(f,g:\, \mathbb{R}\to\mathbb{R}\) of the functional equation \(x+g[y+f(x)]=y+g[x+f(y)]\:(x,y\in\mathbb{R})\) are offered that satisfy also \(f(0)=g(0)=0\). [In Lemma 2, while not completely clear, a relation seems to be offered between differentiable solutions that satisfy the last condition and those that do not]. The last part of the paper contains ``a sketch of a very similar proof'' for the nonexistence of differentiable solutions in the case \(g=f,\) even without assuming \(f(0)=0.\)