Metric Spaces 2000, Lecture 24

The familiar Intermediate Value Theorem of elementary calculus says that if a real valued function f is continuous on the interval [a, b] ⊆ R then it takes each value between f(a) and f(b). As our next result shows, the critical fact is that the domain of f , the interval [a, b], is a connected space, for the theorem generalizes to real-valued functions on any connected space. The Intermediate Value Theorem. Suppose that f :X → R is continuous, where X is a nonempty connected space, and let a, b ∈ X. If y ∈ R and f(a) ≤ y ≤ f(b) then there is an x ∈ X such that f(x) = y.