CONTINUOUS ON RAYS SOLUTIONS OF A GOŁA̧B–SCHINZEL TYPE EQUATION

We show that if the pair $(f,g)$ of functions mapping a linear space $X$ over the field $\mathbb{K}=\mathbb{R}\text{ or }\mathbb{C}$ into $\mathbb{K}$ satisfies the composite equation $$\begin{eqnarray}f(x+g(x)y)=f(x)f(y)\quad \text{for }x,y\in X\end{eqnarray}$$ and $f$ is nonconstant, then the continuity on rays of $f$ implies the same property for $g$. Applying this result, we determine the solutions of the equation.