Definition:Inverse Tangent/Real/Arctangent

From Nature of the Tangent Function, we have that $$\tan x$$ is continuous and strictly increasing on the interval $$\left({-\frac \pi 2 \, . \, . \, \frac \pi 2}\right)$$.

From the same source, we also have that:
 * $$\tan x \to + \infty$$ as $$x \to \frac \pi 2 ^-$$;
 * $$\tan x \to - \infty$$ as $$x \to -\frac \pi 2 ^+$$;

Let $$g: \left({-\frac \pi 2 \, . \, . \, \frac \pi 2}\right) \to \mathbb{R}$$ be the restriction of $$\tan x$$ to $$\left({-\frac \pi 2 \, . \, . \, \frac \pi 2}\right)$$.

Thus from Inverse of Strictly Monotone Function, $$g \left({x}\right)$$ admits an inverse function, which will be continuous and strictly increasing on $$\mathbb{R}$$.

This function is called arctangent of $$x$$ and is written $$\arctan x$$.

Thus:
 * The domain of $$\arctan x$$ is $$\mathbb{R}$$;
 * The image of $$\arctan x$$ is $$\left({-\frac \pi 2 \, . \, . \, \frac \pi 2}\right)$$.

Caution
There exists the a popular but misleading notation $$\tan^{-1} x$$, which is supposed to denote the "inverse tangent function".

However, note that as $$\tan x$$ is not an injection, it does not have an inverse.

The $$\arctan$$ function as defined here has a well-specified image which (to a certain extent) is arbitrarily chosen for convenience.

Therefore it is preferred to the notation $$\tan^{-1} x$$, which (as pointed out) can be confusing and misleading.