Proposition 1

The functions xy and $1/x$, defined respectively on $\mathbf{Q} \times \mathbf{Q}$ and $\mathbf{Q}^*$, can be extended by continuity to $\mathbf{R} \times \mathbf{R}$ and $\mathbf{R}^*$ respectively, and define a field structure on $\mathbf{R}$. Endowed with this structure, $\mathbf{R}$ is called the field of real numbers.

All the properties of topological fields established in § 6 of Chapter III are of course applicable; in particular, every rational function of $n$ real variables, with real coefficients, is continuous at every point of $\mathbf{R}^n$ where its denominator does not vanish.