# Point values and continuity of real functions

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

Continuity on the whole real domain requires a defined center value agreeing with the two-sided limit; extension and original-function continuity are distinct.

mko-continuity · definition · v0.10.0



## Mathematical statement

For f:R→R, continuity at a means that every ε>0 admits δ>0 such that |f(x)−f(a)|<ε whenever |x−a|<δ. Equivalently, its punctured two-sided limit equals the defined f(a). In the two-affine-branch model with explicit center value V, continuity at a holds exactly when left limit=right limit=V.





## Representative formula (exact TeX)

C(f,a)





## Assumptions


- The whole real domain and center value are explicit; an undefined center is not automatically a discontinuity of a function declared on all R.
- Affine point-value questions compute ma+b. Branch questions compute left_m*a+left_b and right_m*a+right_b, with V supplied independently.
- For a continuous full-real extension of the removable quotient, the unique center value is 2a; before extension the original quotient remains defined only on R∖{a}.





## Symbol scope


- C — continuity predicate at a point for a whole-real-domain function
- f — specified total real function or proposed extension
- a — specified real center





## Intuition, definition, and misconceptions

Equal left and right limits alone are insufficient: they must also equal the center value. Agreement of the center with just one unequal side is insufficient as well.

An extension question asks how to define a new total real function, not whether the original punctured quotient already has a center value. Removability does not eliminate the need to define it.

The exact defect pair (left limit−V,right limit−V) shows the mismatch; both must be zero in this model. Common output translation, or slope changes preserving the same side limits, does not create a new defect calculation.

Common errors include ignoring the point value when one-sided limits agree, or claiming center continuity for an unextended quotient.





## Adopted derivation guide

[Continuity: centre values of branch functions and the unique continuous extension across a removable singularity](https://ocme.evemisslab.com/en/derivations/derivation-mko-continuity-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function limits on punctured real domains](https://ocme.evemisslab.com/en/mko/mko-function-limit/) — First check the two-sided limit.
- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — A center value is part of the total-function declaration.





## Verification boundary

Declared branches and extension conditions can be recomputed; general continuity theory and formal Evidence have not been established.

not_formalized

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





## Next obligation

Bind pinned ContinuousAt, the full real domain, and point extension, retaining the distinction from punctured limits and undefined points.





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