Exercise 5, § 3

If we consider $\mathbf{R}^2$ as embedded in $\tilde{\mathbf{R}} \times \tilde{\mathbf{R}}$, and $\mathbf{R}$ as embedded in $\tilde{\mathbf{R}}$, show that the mapping $(x, y) \to x + y$ of $\mathbf{R}^2$ into $\mathbf{R}$ can be extended by continuity to the points $(\infty, a)$ and $(a, \infty)$ of $\tilde{\mathbf{R}} \times \tilde{\mathbf{R}}$ for all finite values of $a$, but that it cannot be extended by continuity to $(\infty, \infty)$. If we consider $\mathbf{R}^2$ as embedded in $\mathbf{P}_2$ ($\mathbf{R}$ still considered as embedded in $\tilde{\mathbf{R}}$), $x + y$ can be extended by continuity to all points of the line at infinity except the point with homogeneous coordinates $(1, -1, 0)$.

State and prove the analogues of these results for the product $xy$.