Exercise 4, § 1

Show that, on the set $\mathscr{C}(\mathbf{R}; \mathbf{R})$ of finite continuous real-valued functions defined on $\mathbf{R}$, the topology of uniform convergence with respect to the additive uniformity of $\mathbf{R}$ is not the same as the topology of uniform convergence with respect to the uniformity induced on $\mathbf{R}$ by the (unique) uniformity of the extended line $\overline{\mathbf{R}}$ (although the topologies induced by these two uniformities on $\mathbf{R}$ are the same).