Definition:Bounded

Ordered Set
Let $$\left({S; \preceq}\right)$$ be a poset.

Let $$T \subseteq S$$ be both bounded below and bounded above in $$S$$.

Then $$T$$ is bounded in $$S$$.

Mapping
Let $$\left({T; \preceq}\right)$$ be a poset.

Let $$f: S \to T$$ be a mapping.

Let the range of $$f$$ be bounded.

Then $$f$$ is defined as being bounded.

That is, $$f$$ is bounded if it is both bounded above and bounded below.

Real Function
The definition for bounded as applied to a real function is the same as for a mapping.

Note that it follows from Bounded Set of Real Numbers‎ that $$f$$ is bounded on a set $$S$$ iff $$\exists K \in \mathbb{R}: \forall x \in S: \left|{f \left({x}\right)}\right| \le K$$.