# Pointwise convergence of functions

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

A function sequence on one domain converges pointwise when its value sequence converges at each fixed x; the witness N may depend on x and epsilon.

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



## Mathematical statement

This slice uses a complete finite domain and the symbolic tail f_n(x)=f(x)+c_x/(n+s). Supply an exact epsilon witness for each x. A maximum of the finitely many witnesses gives uniform convergence for this finite model, not for an arbitrary infinite domain. A finite prefix is insufficient.





## Representative formula (exact TeX)

P(F)





## Assumptions


- All f_n and f use the same explicitly specified finite domain and codomain.
- Epsilon is a positive rational and the tail certificate holds for every n≥N.





## Symbol scope


- P — pointwise convergence
- F — function sequence





## Intuition, definition, and misconceptions

N may depend on both x and epsilon.

A finite domain permits the finite maximum of all pointwise witnesses.

A finite prefix does not prove convergence, and the finite-domain implication does not extend automatically to infinite domains.

Do not exchange quantifiers to infer uniform convergence in general, or infer a limit from finitely many terms.





## Adopted derivation guide

[Pointwise tail certificates on a finite domain](https://ocme.evemisslab.com/en/derivations/derivation-mko-pointwise-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


- [Sequence](https://ocme.evemisslab.com/en/mko/mko-sequence/) — n indexes the functions.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — The domain and codomain are shared.
- [Sequence limits and exact tail cutoffs](https://ocme.evemisslab.com/en/mko/mko-sequence-limit/) — Each fixed point supplies a value sequence.
- [Quantifiers and explicit domains](https://ocme.evemisslab.com/en/mko/mko-quantifier/) — For every x and epsilon there is a witness N(x,epsilon).





## Verification boundary

Only explicit finite domains and symbolic rational tails are handled.

not_formalized

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





## Next obligation

Add uniform convergence and conditions for interchanging 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/pointwise-convergence-of-functions.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-pointwise-convergence-of-functions/index.md)