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## Definition: Bounded and Unbounded Functions

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

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 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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