Alphabeta MathHow 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: semicontinuity implies continuity on a compact set
Statement
False claim: an upper or lower semicontinuous real-valued function on a compact Euclidean set must be continuous.
Facts & Assumptions
Given: The compact interval .
On , the characteristic function of is upper semicontinuous but discontinuous at zero, and its negative is lower semicontinuous but discontinuous there (Characteristic functions of open and closed sets are one-sided semicontinuous).
Refutation
The first function in [L1] satisfies upper semicontinuity on a compact domain and fails continuity.
Its negative in [L1] separately satisfies lower semicontinuity and fails continuity on the same compact domain. Thus neither one-sided notion implies continuity.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
4 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
- D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lectures 2 and 4 (standard reference, not scraped)