Proposition 8

The function xy can be extended by continuity to the set $\overline{\mathbf{R}}^* \times \overline{\mathbf{R}}^*$ according to the formulae

(3)

We leave the proof to the reader; it is analogous to the proof of Proposition 7.

Likewise, we see that $xy$ has no limit at the points $(0, +\infty), (+\infty, 0), (0, -\infty), (-\infty, 0)$ of $\overline{\mathbf{R}} \times \overline{\mathbf{R}}$.

The function $xy$ is a law of composition on $\overline{\mathbf{R}}^*$ which extends the law of multiplication on $\mathbf{R}$; this law is associative and commutative (principle of extension of identities); it has 1 as identity element; and the non-regular elements in $\overline{\mathbf{R}}^*$ are $+\infty$ and $-\infty$.

If $x \leq y$ and $z > 0$ we have $xz \leq yz$ whenever both sides of this inequality are defined. If the product $xy$ is defined, then so is $|x| \cdot |y|$ and we have $|xy| = |x| \cdot |y|$.

Finally, the distributivity formula

is still valid, by virtue of the principle of extension of identities, whenever all the operations which figure on either side are defined.

Note that it can happen that the left-hand side of (4) is defined while the right-hand side is not: for example, consider the case where $x = +\infty$, $y = 2$ and $z = -1$. The distributivity formula should therefore be used with caution in $\overline{\mathbf{R}}$.