intermediate value theorem
If is a real-valued continuous function on the interval ,and and are points with such that ,then for every strictly between and there is a such that .
Bolzano’s theorem is a special case of this.
The theorem can be generalized as follows:If is a real-valued continuous functionon a connected topological space
,and with ,then for every between and there is a such that .(However, this “generalization
” is essentially trivial,and in order to derive the intermediate value theorem from itone must first establish the less trivial fact that is connnected.)This result remains trueif the codomain is an arbitrary ordered set with its order topology;see the entryproof of generalized intermediate value theorem (http://planetmath.org/ProofOfGeneralizedIntermediateValueTheorem)for a proof.