# Graphs of real-valued functions

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

In a fixed Cartesian coordinate system, the graph of a real-valued function is the set of ordered pairs of every permitted input with its unique output.

mko-function-graph · definition · v0.10.0



## Mathematical statement

For D⊆R and a total function f:D→R, define G_f={(x,f(x)):x∈D} in a fixed planar Cartesian coordinate system. An ordered pair (u,v) belongs to G_f exactly when u∈D and v=f(u).





## Representative formula (exact TeX)

P(x,f(x))





## Assumptions


- Fix the coordinate system and ordered real coordinates, and declare the domain D first.
- f has exactly one real output for every input in D. If a finite table declares a codomain B, every output must also belong to B.





## Symbol scope


- P — notation for a graph point
- x — domain input
- f — total real-valued function





## Intuition, definition, and misconceptions

Membership checks both u∈D and v=f(u). Even when an algebraic formula can be evaluated outside D, the resulting point is outside the graph of the declared function.

A finite graph can be sorted by first coordinate, with repeated identical pairs listed once. Given D and B, check exactly one output in B for each input in D and exclude points with inputs outside D. This validates a table for the specified total function.

The vertical-line test checks at most one output per input; by itself it does not check totality on the declared domain D. A sketch supplies no exact coordinates, and one point or finite data do not automatically establish an inverse, surjectivity, or a global formula.

Common errors include checking y=f(x) while omitting the domain, or treating at most one graph point on each vertical line as exactly one output at every declared input.





## Adopted derivation guide

[Graphs of real-valued functions: membership tests and verifying a finite table as a total function](https://ocme.evemisslab.com/en/derivations/derivation-mko-function-graph-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/) — Every permitted input must have a unique output.
- [Coordinate plane](https://ocme.evemisslab.com/en/mko/mko-coordinate-plane/) — Interpret the two graph coordinates in the fixed coordinate system.





## Verification boundary

The explicit finite graphs and integer affine inputs in this pulse can be checked exactly; the general graph-set and geometric-representation bridge awaits formalization.

not_formalized

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





## Next obligation

Bind Set.graph, real product coordinates, domain restrictions, and finite total-function table semantics.





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