Let D be a closed connected domain and let P, Q ∈ D. The Intermediate Value Theorem (IVT) states that if ƒ is continuous on D, then ƒ(x, y) takes on every value between ƒ(P) and ƒ (Q) at some point in D.

(a) Show, by constructing a counterexample, that the IVT is false if D is not connected.

(b) Prove the IVT as follows: Let r(t) be a path such that r(0) = P and r(1) = Q (such a path exists because D is connected). Apply the IVT in one variable to the composite function ƒ(r(t)).