Let \(X\) be a set]. A function \(f:X\to \mathbb R\), i.e. a function mapping this set to the set of real numbers is called

**bounded above**, if \(f(X)\) is bounded above.**bounded below**, if \(f(X)\) is bounded below.**bounded**, if \(f(X)\) is both, bounded below and bounded above.

If \(f(X)\) is not bounded, (i.e. not bounded above or it is not bounded below or neither bounded above nor bounded below), we call \(f\) **unbounded**.

^{1} Please note that $X$ does not necessarily have to be a subset of real numbers $\mathbb R$. The concept of bounded functions can be defined more generally for any kind sets. The only important detail is that the image set of the function is a subset of $\mathbb R$.

| | | | | created: 2014-02-21 20:55:03 | modified: 2020-01-24 08:57:10 | by: *bookofproofs* | references: [581]

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[581] **Forster Otto**: “Analysis 1, Differential- und Integralrechnung einer Veränderlichen”, Vieweg Studium, 1983