# 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.

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

Github: ### References

#### Bibliography

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

#### Footnotes

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 $\mathbb R$.