Exercise 1, § 1

\* a) Let $f$ be a homomorphism of the additive group $\mathbf{R}$ into itself. Show that, if the graph of $f$ is not dense in $\mathbf{R}^2$, $f$ is of the form $x \to ax$ [consider the closure in $\mathbf{R}^2$ of the graph of $f$, and use the structure theorem for closed subgroups of $\mathbf{R}^2$ (Chapter VII, § 1, no. 2, Theorem 2)]. \* [Compare with Chapter VI, § 1, Exercise 12 b) and Chapter IV, § 6, Exercise 2.]

b) If the graph of $f$ is dense in $\mathbf{R}^2$, the inverse image of the topology of $\mathbf{R}^2$ under the mapping $x \to (x, f(x))$ is compatible with the group structure of $\mathbf{R}$ and is strictly finer than the usual topology of $\mathbf{R}$. If moreover $f$ is injective, the inverse image under $f$ of the usual topology of $\mathbf{R}$ is compatible with the group structure of $\mathbf{R}$ and is not comparable with the usual topology of $\mathbf{R}$.