# intermediate value theorem

If $f$ is a real-valued continuous function   on the interval $[a,b]$, and $x_{1}$ and $x_{2}$ are points with $a\leq x_{1} such that $f(x_{1})\neq f(x_{2})$, 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 $f(x_{1})\neq f(x_{2})$, 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 (http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem) for a proof.

 Title intermediate value theorem Canonical name IntermediateValueTheorem Date of creation 2013-03-22 11:51:29 Last modified on 2013-03-22 11:51:29 Owner yark (2760) Last modified by yark (2760) Numerical id 15 Author yark (2760) Entry type Theorem Classification msc 26A06 Classification msc 70F25 Classification msc 17B50 Classification msc 81-00 Related topic RollesTheorem Related topic MeanValueTheorem Related topic Continuous