If $f$ is a real-valued continuous function on the interval $[a,b]$ , and $x_1$ and $x_2$ are points with $a\le x_1<x_2\le b$ such that $f(x_1)\ne f(x_2)$ , then for every $y$ strictly between $f(x_1)$ and $f(x_2)$ there is a $c\in(x_1,x_2)$ such that $f(c)=y$ .

Bolzano's theorem is a special case of this.

The theorem can be generalized as follows: If $f$ is a real-valued continuous function on a connected topological space $X$ , and $x_1, x_2 \in X$ with $f(x_1) \ne f(x_2)$ , then for every $y$ between $f(x_1)$ and $f(x_2)$ there is a $\xi \in X$ such that $f(\xi) = y$ . (However, this ``generalization'' is essentially trivial, and in order to derive the intermediate value theorem from it one must first establish the less trivial fact that $[a,b]$ is connnected.) This result remains true if the codomain is an arbitrary ordered set with its order topology; see the entry proof of generalized intermediate value theorem for a proof.