Intermediate value theorem

Intermediate value theorem: Let $f:[a,b]\to\mathbb{R}$ be continuous on $[a,b]$ . If $y$ is any number between $f(a)$ and $f(b)$ (that is, $\min\{f(a),f(b)\}\le y\le \max\{f(a),f(b)\}$ ), then there exists $c\in[a,b]$ such that

f(c)=y.

Remarks

This is one of the basic consequences of being a continuous map on an interval. A notable application is Darboux's theorem, which shows that derivatives also satisfy an intermediate value property.

Proof from completeness

It suffices to treat $f(a)<y<f(b)$ ; endpoint equalities are immediate, and replacing $f$ by $-f$ handles the reversed case. Let $c$ be the supremum of $S=\{x\in[a,b]:f(x)\le y\}$ . The set is nonempty and bounded. If $f(c)<y$ , continuity provides points of $S$ strictly to the right of $c$ , a contradiction. If $f(c)>y$ , continuity excludes $S$ from a left neighborhood of $c$ , also contradicting the definition of supremum. Hence $f(c)=y$ .