Continuous on rays solutions of an equation of the Goła̧b–Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14], and J. Brzdȩk (2003) [11].