**Definition (Bounded Function)** Given a [[Function]] $f: E \to \mathbb{R}$. - The function is *bounded above (on $E$*) if there exists an $M \in \mathbb{R}$ such that $f(x)\leq M \quad \forall x\in E$ - The function is *bounded below (on $E$)* if there exists $m \in \mathbb{R}$ such that $f(x) \geq m \quad \forall x \in E.$ - We say the function is *bounded (on $E$)* if it is bounded above and below. - If $f$ is bounded above on $E$ (and $E$ is non-empty) then the set $f(E) = \{ f(x) \mid x \in E \}$ is non-empty and bounded above, so has a supremum (by [[Real numbers|LUBA]]) which we denote $\underset{x\in E}{\sup} f(x) = \sup {f(x) \mid x \in E}$ **Note** that we say $f$ is *bounded* if the [[Image of a set under a function|image]] of $E$ under $f$ is a [[Bound of Set of Reals|bounded set]].