DefinitionDefinition: AI-adaptedProof: Not applicableSession-authored (Fable 5 assisted)judge 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.

Upper and lower semicontinuity on subsets of $R^{n}$

Definition

Let $n \geq 1$ , let $A \subseteq R^{n}$ , let $f : A \to R$ , and let $a \in A$ . Using the Euclidean metric and its balls ( $R^{n}$ as the set of functions $n \to R$ , and $d_{1}$ , $d_{2}$ , $d_{\infty}$ are metrics on it, Open ball, closed ball and sphere in a metric space), $f$ is upper semicontinuous at $a$ when for every $ε > 0$ there is $δ > 0$ such that $f (x) < f (a) + ε$ for every $x \in A \cap B_{2} (a, δ)$ .

The function is lower semicontinuous at $a$ when for every $ε > 0$ there is $δ > 0$ such that

$f (a) - ε < f (x) (x \in A \cap B_{2} (a, δ)) .$

It is upper or lower semicontinuous on $A$ when the corresponding condition holds at every point of $A$ . These are relative notions on $A$ ; when $A$ is empty, each on-set condition is vacuous. A continuous function (Continuity of a map between metric spaces, at a point and globally, in the $ε$ - $δ$ form) satisfies both conditions.

Depends on

Used by

Dependency tree · two levels

23 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)

Upper and lower semicontinuity on subsets of RnRn