# A uniform limit of continuous functions is continuous

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

Continuous functions on one common domain have a continuous limit when convergence is uniform on that domain.

mko-uniform-limit-continuity-theorem · theorem · v0.10.0



## Mathematical statement

Every approximating f_n is continuous on the same declared domain and sup_x|f_n(x)−f(x)| tends to0. Exact families use affine perturbations on[0,1], exact supremum errors and epsilon–N certificates. An epsilon/3 transfer fixes one continuous approximant f_N and passes its continuity to f. Pointwise-only powers, uniform sequences whose terms are discontinuous and finite-grid agreement do not supply both theorem hypotheses.





## Representative formula (exact TeX)

U(F,f,e)=b





## Assumptions


- All functions share a declared domain and codomain.
- The uniform N depends on epsilon but not on x; approximant continuity is a separate premise.





## Symbol scope


- U — uniform-limit theorem operator
- F — continuous-function sequence
- f — uniform-limit function
- e — positive tolerance
- b — continuity conclusion





## Intuition, definition, and misconceptions

Supremum error and epsilon–N quantifiers precede the conclusion.

The epsilon/3 transfer fixes one continuous approximant and does not require a common delta for all terms.

Pointwise convergence and finite samples are insufficient.

Pointwise or finite-grid closeness is not uniform convergence, and continuity of the approximants cannot be omitted.





## Adopted derivation guide

[An epsilon/3 transfer through one fixed approximant](https://ocme.evemisslab.com/en/derivations/derivation-mko-uniform-limit-continuity-theorem-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Uniform convergence of functions](https://ocme.evemisslab.com/en/mko/mko-uniform-convergence-of-functions/) — Use the essential convergence hypothesis.
- [Point values and continuity of real functions](https://ocme.evemisslab.com/en/mko/mko-continuity/) — Separate term continuity and the limit conclusion.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Declare the common domain and codomain.
- [Supremum and infimum](https://ocme.evemisslab.com/en/mko/mko-supremum-and-infimum/) — Control the error over the entire domain.





## Verification boundary

The affine exact families check quantifiers and bounds; general metric/topological versions still need their own proof.

not_formalized

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





## Next obligation

Add the extra hypotheses for interchanging limits with integration or differentiation.





## 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-limit-continuity-theorem.json) · [Plain Markdown](https://ocme.evemisslab.com/en/mko/mko-uniform-limit-continuity-theorem/index.md)