Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: a convex function on a convex set is continuous

Statement

False claim: every convex real-valued function on a convex Euclidean domain is continuous at every domain point.

Facts & Assumptions

Given: No openness assumption is imposed.

[L1]

There is a function on [0,1] that is convex on [0,1] but is not continuous at 0 (A convex function on [0,1] that is discontinuous at the boundary).

Refutation

technique · direct
1.1

The interval in [L1] is convex and the displayed function is convex on it.

L1
2.1

The same function fails continuity at the boundary point zero by [L1], so it refutes the claim. Openness of the domain, or restriction to its interior, is essential.

step 1.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources