DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-27

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-semicontinuous regularization

Definition

Let $Ω \subseteq C$ be open and let $v : Ω \to [- \infty, \infty)$ be any function. Its upper-semicontinuous regularization is $v^{*} (z) : = {lim sup}_{w \to z} v (w) = \lim_{ρ ↓ 0} \sup {v (w) : ∣ w - z ∣ < ρ, w \in Ω} .$

Remarks

The function $v^{*}$ is upper semicontinuous by construction and satisfies $v \leq v^{*}$ . It is the least upper-semicontinuous majorant of $v$ : any upper-semicontinuous $g$ with $v \leq g$ also satisfies $v^{*} \leq g$ .

Used by

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Sheldon Axler, Paul Bourdon, and Wade Ramey, Harmonic Function Theory, 2nd ed. (standard reference, not scraped)