Proposition 2

Let X be a locally compact space in which a locally compact group H acts on the right, continuously and properly, by $(x, \xi) \mapsto x\xi$. Assume that X/H is paracompact. Let g be a continuous representation of H in $\mathbf{R}^n$. Then there exists a continuous mapping f of X into $\mathbf{R}^n$ such that $f(x\xi) = f(x) + g(\xi)$ for all $x \in X$ and $\xi \in H$.

One reduces immediately to the case that $n = 1$. Since the additive group $\mathbf{R}$ is isomorphic to the multiplicative group $\mathbf{R}_+^*$, the proposition is then an immediate consequence of Prop. 7 of §2, No. 4.