# Multivariable real-valued function

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

A multivariable real-valued function maps inputs in an explicitly declared real-vector domain to real numbers; coordinate roles and domain are function data.

mko-multivariable-real-function · definition · v0.10.0



## Mathematical statement

This entry initially restricts attention to explicit slices of bivariate real polynomials f:R²→R, with coefficients ordered as [1,x,y,x²,x*y,y²]. Function qualification requires a declared domain and coordinate roles; continuity, differentiability and vector-valued codomains are not automatically asserted.





## Representative formula (exact TeX)

f(x,y)





## Assumptions


- Exercises use six integer coefficients and integer coordinate points.
- Rectangle-domain audits check only an explicitly declared closed rectangle. Although polynomial expressions can be globally defined, the exercise still follows its declared domain.





## Symbol scope


- f — bivariate real-valued function
- x — first coordinate
- y — second coordinate





## Intuition, definition, and misconceptions

A fixed coefficient order is necessary for exact recomputation.

Rectangle-domain membership requires checking the x and y bounds separately.

Fixing y gives only one x-slice; a finite slice table does not determine the full function.

Do not swap x and y coefficients or use a few point values to establish global function equivalence, continuity or differentiability.





## Adopted derivation guide

[Multivariable real-valued function: coefficient order, declared domain, slices and point values of a bivariate quadratic](https://ocme.evemisslab.com/en/derivations/derivation-mko-multivariable-real-function-01/)The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.





## Prerequisites


- [Function mapping](https://ocme.evemisslab.com/en/mko/mko-function-mapping/) — Each admissible input receives exactly one real output.
- [Plane vectors in a fixed basis](https://ocme.evemisslab.com/en/mko/mko-vector/) — A multivariable input can be viewed as a real coordinate vector.
- [Coordinate plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) — Bivariate inputs use ordered coordinates.





## Verification boundary

Only R²→R polynomial slices are checked; partial derivatives and general regularity are not established.

not_formalized

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





## Next obligation

Add partial derivatives, gradients and PDEs.





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