Definition: Extended Real Numbers

The set $\mathbb R\cup \{-\infty,+\infty\},$ i.e the set of real numbers extended by the improper real numbers $-\infty$ minus infinity) and $+\infty$ plus infinity), is called the extended real numbers and denoted by $\overline{\mathbb R}.$

Extending the order relation for real numbers, we define $-\infty < x < +\infty$ for all $x\in\mathbb R.$

Definitions: 1 2 3 4 5
Lemmas: 6 7

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