Graphs of real-valued functions

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

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Function mapping — Every permitted input must have a unique output.
• 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.