Boundedness (Real-Valued Functions)#

Definition: Boundedness

Let \(f: \mathcal{D} \to \mathbb{R}\) be a real-valued function and let \(S \subseteq \mathcal{D}\).

We say that \(f\) is:

• bounded above on \(S\) if there exists some \(u \in \mathbb{R}\) such that \(f(x) \le u\) for all \(x \in S\). Any such \(u\) is called an upper bound of \(f\) on \(S\).
• bounded below on \(S\) if there exists some \(l \in \mathbb{R}\) such that \(f(x) \ge l\) for all \(x \in S\). Any such \(l\) is called a lower bound of \(f\) on \(S\).
• bounded on \(S\) if it is both bounded below and bounded above.