intermediate value theorem
Bolzano’s theorem is a special case of this.
The theorem can be generalized as follows: If is a real-valued continuous function on 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 it one must first establish the less trivial fact that 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|
|Date of creation||2013-03-22 11:51:29|
|Last modified on||2013-03-22 11:51:29|
|Last modified by||yark (2760)|