Answer Happy

Let f: A → R be a continuous function. If E CA is connected, then f(E) is connected. Now, by using this theorem, prove the Intermediate Value Theorem. Namely, prove that if f [a, b] → R is continuous and LER is such that f(a) < L < f(b) or f(a) > L > f(b), then there exists c = (a, b) such that f(c) = L.
Proof. Intending to use the characterization of connected sets in Theorem 3.4.6, let f(E) = AUB where A and B are disjoint and nonempty. Our goal is to produce a sequence contained in one of these sets that converges to a limit in the other. Let C = {r € E: f(x) = A} and D = {r € E : f(x) € B}. The sets C and D are called the preimages of A and B, respectively. Using the properties of A and B, it is straightforward to check that C and D are nonempty and disjoint and satisfy E = CUD. Now, we are assuming E is a connected set, so by Theorem 3.4.6, there exists a sequence (₁) contained in one of C or D with x = lim ₁ contained in the other. Finally, because f is continuous at x, we get f(x) = lim f(zn). Thus, it follows that f(n) is a convergent sequence contained in either A or B while the limit f(x) is an element the other. With another nod to Theorem 3.4.6, the proof is complete.