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].