Concept preview

What you will be able to do

1. 01State the Intermediate Value Theorem precisely, including its continuity hypothesis, and explain that it guarantees the EXISTENCE of a solution, not its exact value.
2. 02Apply the IVT to guarantee a root or a target output on a closed interval by checking continuity and comparing the endpoint values.
3. 03Recognize when the IVT does NOT apply (a discontinuity in the interval) and avoid over-claiming what the theorem provides.

This lesson (AP topic 1.16) is the payoff of everything you have learned about limits and continuity. The Intermediate Value Theorem, or IVT, is your first existence theorem: a result that guarantees a solution exists even when you cannot compute it exactly. It formalizes an intuition you already trust, that a continuous curve drawn without lifting your pencil cannot get from below a line to above it without crossing the line somewhere.