How can I prove the function $f\left(x\right) = \arctan\left(x\right)$ fulfils these requirements? I was thinking about using supremum/infimum as I know the function is bounded by $-\frac{\pi}{2}\ ,\frac{\pi}{\ 2}$, but how do I show that they're not part of the image?
Arctan is strictly increasing, so its sup and inf are its limits in $\pm \infty$, so $\pm \frac{\pi}{2}$.
But we can't have Arctan $x$ $=\pm\frac{\pi}{2}$, since $\tan \pm \frac{\pi}{2}$ is not defined.