On the measurable solutions of a functional equation

The measurable solutions ${f:mathbb{R}^{3}setminus{0}tomathbb{C}setminus{0}, {rm and}, (t,s)mapsto G(t,s)inmathbb{C}setminus{0},, sinmathbb{R}^{3},, t>|s| >0 }$ of the functional equation $$f(x)f(y)=Gleft(|x|+|y|,x+yright),quad x,yinmathbb{R}^{3}, xtimes yneq 0$$ are considered and it is proved that they are continuous.