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

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.