Proposition 6

*A mapping $f$ of an interval $I$ of $\overline{\mathbf{R}}$ into $\overline{\mathbf{R}}$ is a homeomorphism of $I$ onto $f(I)$ if and only if $f$ is strictly monotonic and continuous on $I$; $f(I)$ is then an interval of $\overline{\mathbf{R}}$.

Finally, the functions $\sup(x, y)$ and $\inf(x, y)$ are continuous on $\overline{\mathbf{R}} \times \overline{\mathbf{R}}$.