A uniform limit of continuous functions is continuous

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

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 — Use the essential convergence hypothesis.
• Point values and continuity of real functions — Separate term continuity and the limit conclusion.
• Function mapping — Declare the common domain and codomain.
• 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.