CorollaryStatement: Literature-sourcedProof: 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.

A convex function on an open convex set is continuous

Statement

Every convex function $f : U \to R$ on an open convex set is continuous on $U$ . The assertion is vacuous when $U$ is empty.

Facts & Assumptions

Given: A convex function on an open convex Euclidean set.

[L1]

Such a function is locally Lipschitz on $U$ (A convex function on an open convex set is locally Lipschitz).

[L2]

Every Lipschitz map is uniformly continuous, and every uniformly continuous map is continuous (Contraction implies Lipschitz implies uniformly continuous implies continuous; every Hölder map is uniformly continuous, and a Lipschitz map on a bounded space is Hölder for every exponent).

Proof

technique · direct

1.1

At each $a \in U$ , [L1] gives a neighbourhood on which the restriction of $f$ is Lipschitz, and [L2] makes that restriction continuous.

L1L2

2.1

Thus $f$ is continuous at every domain point, hence continuous on $U$ ; if $U$ is empty, there is no point to check.

step 1.1∎

Depends on

Used by

Dependency tree · two levels

20 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

S. Boyd and L. Vandenberghe, Convex Optimization, §3.1 (standard reference, not scraped)
D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lecture 4 (standard reference, not scraped)