Definition: Bounded and Unbounded Functions

Let \(X\) be a set1. A function \(f:X\to \mathbb R\), i.e. a function mapping this set to the set of real numbers is called

  1. bounded above, if \(f(X)\) is bounded above.
  2. bounded below, if \(f(X)\) is bounded below.
  3. 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. Proposition: Continuous Real Functions on Closed Intervals Take Maximum and Minimum Values within these Intervals

Corollaries: 1 2 3
Definitions: 4 5 6
Examples: 7
Lemmas: 8
Proofs: 9 10 11 12 13 14 15 16
Propositions: 17 18 19
Subsections: 20

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


  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 $\mathbb R$.