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Boundedness of Real Functions#

Definition: Boundedness

Let \(f: \mathcal{D} \subset \mathbb{R} \to \mathbb{R}\) be a real function, let \(S \subseteq \mathcal{D}\) and let \(r \in \mathbb{R}\).

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 there exists some \(B \in \mathbb{R}_{\ge 0}\) such that \(|f(x)| \le B\) for all \(x \in S\).

If \(f\) is both bounded below and bounded above on \(S\), then we say that \(f\) is bounded on \(S\).