Continuous solutions of a generalization of the Gołab–Schinzel equation II

Summary.

Let $\phi, \psi :\mathbb{R} \to \mathbb{R}$ be given continuous functions such that $|\phi \left( 1 \right)| = |\psi \left( 1 \right)| = 1.$ We determine all continuous solutions $f:\mathbb{R} \to \mathbb{R}$ of the functional equation

$$ f{\left( {x\phi {\left[ {f(y)} \right]} + y\phi {\left[ {f(x)} \right]}} \right)} = f(x)f(y). $$