Intermediate value theorem

Mathematical statement

Fix a<b, the entire interval domain and a continuous polynomial certificate. The target y must lie inclusively between f(a) and f(b), in either orientation. Exact families supply c∈[a,b] with f(c)=y and retain endpoint roots. IVT gives existence, not uniqueness, a rational closed form or convergence of finitely many bisection steps. An explicit discontinuous step can change sign without taking the target value, and finite samples do not prove continuity.

Representative formula (exact TeX)

I(f,a,b,y)=c

Assumptions

• Exact rational polynomials provide verifiable continuous instances.
• Root witnesses, endpoints and bracketing are checked exactly.

Symbol scope

• I — intermediate-value construction
• f — continuous real function
• a — left endpoint
• b — right endpoint
• y — target value
• c — intermediate point

Intuition, definition, and misconceptions

Qualify interval, continuity and bracket completely.

An exact root witness is not uniqueness.

Discontinuous counterexamples protect the continuity premise.

Endpoint sign change without continuity does not imply a root.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Point values and continuity of real functions — Retain the essential hypothesis.
• Real number — Use real intervals and values.
• Usual order on the real numbers — Specify the between relation.
• Function mapping — Declare domain and codomain.
• Supremum and infimum — Retain the general proof background.

Verification boundary

Polynomial instances do not replace a general IVT proof; finite bisection is not exact convergence. A fixed uniqueness/closed-form nonclaim does not deny a particular supplied rational root.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add root-finding convergence boundaries.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.