Published Jul 02, 2021
[
Theorem 4.5.1 (Intermediate Value Theorem). Let \(f: [a, b] \to \mathbb{R}\) be continuous. If \(L\) is a real number satisfying \(f(a) < L < f(b)\) or \(f(a) > L > f(b)\), then there exists a point \(c \in (a, b)\) where \(f(c) = L\)
Theorem 4.5.2 (Preservation of Connected Sets). Let \(f : G \to \mathbb{R}\) be continuous. If \(E \subseteq G\) is connected, then \(f(E)\) is connected as well.
Definition 4.5.3 (Intermediate Value Property). A function \(f\) has the intermediate value property on an interval \([a, b]\) if for all \(x < y\) in \([a, b]\) and all \(L\) between \(f(x)\) and \(f(y)\), it is always possible to find a point \(c \in (x, y)\) where \(f(c) = L\).