# Uniform convergence of functions

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

Uniform convergence requires one witness N(epsilon) controlling every point of the domain simultaneously.

mko-uniform-convergence-of-functions · definition · v0.10.0



## Mathematical statement

Use a finite domain and symbolic rational tails. The maximum of all pointwise witnesses gives a uniform witness on this complete finite domain, not on arbitrary infinite domains. Check exact supremum gaps and the all-n tail; a finite prefix is insufficient.





## Representative formula (exact TeX)

U(F)





## Assumptions


- The finite domain and codomain are shared, and epsilon is a positive rational.
- One N controls every point and every n≥N.





## Symbol scope


- U — uniform convergence
- F — function sequence





## Intuition, definition, and misconceptions

N must not depend on x.

A finite domain permits the maximum of pointwise witnesses.

A finite prefix does not prove the tail, and the finite-domain argument does not extend automatically to an infinite domain.

Separate pointwise witnesses are not a common witness on an arbitrary domain, and finitely tested points cannot certify an infinite-domain supremum.





## Adopted derivation guide

[Exact supremum error on a three-point domain](https://ocme.evemisslab.com/en/derivations/derivation-mko-uniform-convergence-of-functions-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Pointwise convergence of functions](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/) — Finite-domain pointwise witnesses supply the maximum construction.
- [Metric space](https://ocme.evemisslab.com/en/mko/mko-metric-space/) — Errors are measured in the specified metric.
- [Absolute value](https://ocme.evemisslab.com/en/mko/mko-absolute-value/) — Real-valued supremum errors use absolute values.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — Uniformity fixes the quantifier order.





## Verification boundary

The universal certificate is limited to the explicit finite-domain symbolic tail.

not_formalized

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





## Next obligation

Add continuity of uniform limits.





## 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/uniform-convergence-of-functions.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/index.md)