The Arctangent: Continuity, Monotonicity, Derivative, Special Values and Oddness

In the setting of The Real Line: Standing Notation and Background for Calculus on Intervals, let $\arctan:\mathbb{R}\to(-\pi/2,\pi/2)$ be the arctangent and let $\pi$ be the number of The Number Pi (i).

(i) Continuity and monotonicity. $\arctan$ is strictly increasing, that is, $\arctan x<\arctan y$ for all $x,y\in\mathbb{R}$ with $x<y$, and $\arctan$ is continuous on $\mathbb{R}$.

(ii) Derivative. $\arctan$, read as a function $\mathbb{R}\to\mathbb{R}$, is differentiable, and for every $y\in\mathbb{R}$

$$\arctan'(y)=\frac{1}{1+y^{2}}.$$

(iii) Special values.

$$\arctan 0=0,\qquad \arctan 1=\frac{\pi}{4}.$$

(iv) Oddness. For every $y\in\mathbb{R}$,

$$\arctan(-y)=-\arctan y .$$