Proposition: Inverse Tangent of a Real Variable

The tangent of a real variable $\mathbb R\to\mathbb R,~x\to \tan(x)$, is invertible on all real intervals $x\in[-\pi/2+k\pi,\pi/2+k\pi],$ where $\pi$ denotes the $\pi$ constant, and $k\in\mathbb Z$ denotes an integer. Its inverse function $\arctan$, called the inverse tangent, is continuous, strictly monotonically increasing, and defined by \[\arctan:\mathbb R\to\mathbb R.\]

Since the above proposition holds for all $k\in\mathbb Z$, the special case $k=0$ is called the principal branch of $\arctan$. The inverse tangent has a graph shown in the following figure:

Proofs: 1

  1. Proposition: Inverse Tangent and Complex Exponential Function
  2. Proposition: Derivative of the Inverse Tangent
  3. Proposition: Integral of the Inverse Tangent

Definitions: 1
Propositions: 2 3 4

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  1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983