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Intermediate Value Theorem

Middle School

Definition

If continuous f on [a,b] and k is between f(a) and f(b), there exists c in (a,b) with f(c) = k.

If f(a) < k < f(b), then ∃c ∈ (a,b) with f(c) = k

Example 1

Show f(x) = x³ - x - 1 has a root in [1,2].

Discovered by: Bernard Bolzano (1817)

Bolzano gave first rigorous proof. Seems obvious intuitively but requires completeness of reals.

Bisection method

Equilibrium existence

Continuous change analysis