Corollary 1

Let $V$ be a neighbourhood of $o$ in $\mathbf{R}^n$ and let $f$ be a continuous mapping of $V$ into a topological group $G$ such that

for each pair of points $x, y$ such that $x \in V, y \in V$ and $x + y \in V$. Then there exists a unique continuous homomorphism of $\mathbf{R}^n$ into $G$ which agrees with $f$ at all points of a neighbourhood $W$ of $o$.

Take $W$ to be an open box with centre $o$, contained in $V$, and apply Proposition 2 to $W$.

We shall see later that this property of $\mathbf{R}^n$ extends to a larger class of topological groups, the "simply connected" groups.