Bounds of Real Functions

Boundedness

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 $S$.

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