Intermediate Value Theorem

Intermediate Value Theorem (IVT)

Let $a < b \in R$ where $f : [a, b] \to R$ be continuous. Let $y \in R$ satisfies either $f (a) > y > f (b)$ or $f (a) < y < f (b)$ then $y \in f ([a, b])$ .

Proof

Using Continuous Image of Connected Set is Connected, since $[a, b]$ is a connected set, then $f ([a, b])$ is connected. Using Condition for Connected (which we get immediately from the givens), then $y \in f ([a, b])$ immediately (use $a = f (a) < y < f (b) = b$ or vice versa, and since $y \in R$ and in between the $f$ 's then $y \in f ([a, b])$ ).

#todo/school : Do this problem (in the HW) 📅 2024-11-12 ✅ 2024-11-13

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