Remark

The same argument shows, more generally, that if $\varphi$ is a strict morphism of $\mathbf{R}^n$ into a group $G$, whose restriction to a suitable neighbourhood of o is a local isomorphism of $\mathbf{R}^n$ with $G$, then every continuous homomorphism of $\mathbf{R}^n$ into $G$ is of the form $x \mapsto \varphi(u(x))$, where $u$ is a linear mapping of $\mathbf{R}^m$ into $\mathbf{R}^n$.

In the case $m = n = 1$, Proposition 3 gives: