A remark on solutions of a generalization of the addition formulae

Summary.

Let J be a real interval and $H:\mathbb{R}^2 \to \mathbb{R}$. Under some assumptions we determine all continuous functions $g:\mathbb{R} \to J$ and $M:J \to \mathbb{R}$ satisfying the equation

$$ g\left( {x + M\left( {g\left( x \right)} \right)y} \right) = H\left( {g\left( x \right),g\left( y \right)} \right). $$

We also show some consequences of this result.