We show that in the realm of real-valued functions on \({\mathbb {R}}\) and for every real \(a\not =0\) the functional equation \(\begin{aligned}f(x+f(y))=f(x)+f(y)+ay,\end{aligned}\) which appeared 2023 in the problem section of the Mathematical Gazette, always has uncountably many solutions which are discontinuous everywhere. Moreover, if \(a>-1/4\) , \(a\not =0\) , there are exactly two continuous solutions, exactly one if \(a=-1/4\) , and none if \(a<-1/4\) . All of these are \({\mathbb {R}}\) -linear. If \(a=0\) , there are countably infinitely many continuous solutions, all of them affine. A full description of all solutions is given in the case \(a<-1/4\) and \(a=0\) . The latter involves A-periodic functions, where A is an additive subgroup of \({\mathbb {R}}\) . The final section deals with the equation \(\begin{aligned}f(x+f(y))=f(x)+f(y)+ax,\end{aligned}\) which displays for \(a\not =0\) a radically different behavior.
