Let $g:\mathbf{R}\to (-\pi /2,\pi /2)$ be given by $g\left(u\right)$ $=2{\tan }^{-1}\left(e^u\right)-\pi /2,$then $g$ is

odd and is strictly decreasing in $(-\mathrm{\infty },\mathrm{\infty })$

neither even nor odd, but is strictly increasing in  $(-\mathrm{\infty },\mathrm{\infty }).$

even and is strictly increasing in $(0,\mathrm{\infty })$

odd and is strictly increasing in $(-\mathrm{\infty },\mathrm{\infty })$