Intermediate Value Theorem

analysis

Definition

Intermediate Value Theorem

Let $f:[a,b]\to \mathbb{R}$ be a continuous function on the closed interval $[a,b]$. For every value $y$ between $f(a)$ and $f(b)$, there exists $c\in [a,b]$ with $f(c)=y$.

Equivalently, the image of the interval is itself an interval:

$$f\left([a,b]\right)\supseteq \left[\min (f(a),f(b)),\max (f(a),f(b))\right].$$

A horizontal line at any height $y$ between $f(a)$ and $f(b)$ must meet the continuous curve in at least one point $c$.