Intermediate Value Theorem - Generalization

Generalization

The intermediate value theorem can be seen as a consequence of the following two statements from topology:

• If X and Y are topological spaces, f : X → Y is continuous, and X is connected, then f(X) is connected.
• A subset of R is connected if and only if it is an interval.

The intermediate value theorem generalizes in a natural way: Suppose that X is a connected topological space and (Y, <) is a totally ordered set equipped with the order topology, and let f : X → Y be a continuous map. If a and b are two points in X and u is a point in Y lying between f(a) and f(b) with respect to <, then there exists c in X such that f(c) = u. The original theorem is recovered by noting that R is connected and that its natural topology is the order topology.