**Corollary** If $f:[a,b] \to \mathbb{R}$ is continuous then $f([a,b])=[\inf f, \sup f]$. **Proof** We have already shown [[Continuous Image of an Interval is an Interval]]. The [[Extreme Value Theorem]] shows that $\sup_{x\in[a,b]} f(x) = f(x^{*}) \in f([a,b]) \text{ and } \inf_{x\in [a,b]} f(x) = f(x_{*}) \in f([a,b]) $and so $f([a,b])=[f(x_{*}), f(x^{*})]$.