\begin{definition}[Definition:Inverse Cotangent/Real/Arccotangent]
{{:Graph of Arccotangent Function}}
From Shape of Cotangent Function, we have that $\cot x$ is continuous and strictly decreasing on the interval $\openint 0 \pi$.
From the same source, we also have that:
:$\cot x \to + \infty$ as $x \to 0^+$
:$\cot x \to - \infty$ as $x \to \pi^-$
Let $g: \openint 0 \pi \to \R$ be the restriction of $\cot x$ to $\openint 0 \pi$.
Thus from Inverse of Strictly Monotone Function, $\map g x$ admits an inverse function, which will be continuous and strictly decreasing on $\R$.
This function is called '''arccotangent''' of $x$ and is written $\arccot x$.
Thus:
:The domain of $\arccot x$ is $\R$
:The image of $\arccot x$ is $\openint 0 \pi$.
\end{definition}