\begin{definition}[Definition:Inverse Tangent/Real/Arctangent]
{{:Graph of Arctangent Function}}
From Shape of Tangent Function, we have that $\tan x$ is continuous and strictly increasing on the interval $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
From the same source, we also have that:
:$\tan x \to + \infty$ as $x \to \dfrac \pi 2 ^-$
:$\tan x \to - \infty$ as $x \to -\dfrac \pi 2 ^+$
Let $g: \openint {-\dfrac \pi 2} {\dfrac \pi 2} \to \R$ be the restriction of $\tan x$ to $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly increasing on $\R$.
This function is called '''arctangent''' of $x$ and is written $\arctan x$.
Thus:
:The domain of $\arctan x$ is $\R$
:The image of $\arctan x$ is $\openint {-\dfrac \pi 2} {\dfrac \pi 2}$.
\end{definition}