The Arctangent Function

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\tan:(-\pi/2,\pi/2)\to\mathbb{R}$ be the tangent function, with $\pi$ as there.

(i) The arctangent. The arctangent is the inverse

$$\arctan:\mathbb{R}\to(-\pi/2,\pi/2)$$

of the bijection $\tan$ of The Tangent: Derivative, Monotonicity, Bijectivity onto $\mathbb{R}$ and Special Values (iii): for $y\in\mathbb{R}$, $\arctan y$ is the unique $x\in(-\pi/2,\pi/2)$ with $\tan x=y$.