# Intermediate value theorem

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

A continuous real function on a closed interval attains each value between its endpoint values.

mko-intermediate-value-theorem · theorem · v0.10.0



## 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

[A polynomial bracket and an exact intermediate value](https://ocme.evemisslab.com/en/derivations/derivation-mko-intermediate-value-theorem-01/)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](https://ocme.evemisslab.com/en/mko/mko-continuity/) — Retain the essential hypothesis.
- [Real number](https://ocme.evemisslab.com/en/mko/mko-real-number/) — Use real intervals and values.
- [Usual order on the real numbers](https://ocme.evemisslab.com/en/mko/mko-order-relation/) — Specify the between relation.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Declare domain and codomain.
- [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-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.

[Canonical MKO JSON](https://ocme.evemisslab.com/data/mko/intermediate-value-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-intermediate-value-theorem/index.md)