PropositionStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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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.

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Euclidean semicontinuity agrees with the published real-line definition

Statement

Under the standard identification $R^{1} ≅ R$ , a real-valued function on $A \subseteq R$ is upper or lower semicontinuous according to Upper and lower semicontinuity on subsets of $R^{n}$ if and only if it is upper or lower semicontinuous according to Upper and lower semicontinuity of $f : A \to R$ at a point of $A$ and on $A$ .

Facts & Assumptions

Given: A subset $A$ of the real line and a function $f : A \to R$ . The usual real and metric neighbourhoods agree Dictionary: for $A \subseteq R$ with the metric $d (x, y) = ∣ x - y ∣$ , continuity and uniform continuity of $f : A \to R$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $R$ is compact in the open-cover sense of $R$ exactly when it is a compact metric subspace.

[F1]

In the Euclidean definition, $f$ is upper semicontinuous at $a$ when for every $ε > 0$ there is $δ > 0$ such that $f (x) < f (a) + ε$ for every $x \in A \cap B_{2} (a, δ)$ (Upper and lower semicontinuity on subsets of $R^{n}$ ).

[F2]

In the real-line definition, $f$ is upper semicontinuous at $c$ when for every real $ε > 0$ there is a real $δ > 0$ giving the same inequality on $A \cap N_{δ} (c)$ ; the lower clause reverses the one-sided bound (Upper and lower semicontinuity of $f : A \to R$ at a point of $A$ and on $A$ ).

Proof

technique · direct

1.1

Under $R^{1} ≅ R$ , one has $B_{2} (c, δ) = N_{δ} (c)$ for every centre and positive radius. Substitution makes the upper clauses [F1] and [F2] identical in both directions, and the same set identity makes the two lower clauses identical.

F1F2givenalgebra

2.1

Therefore the pointwise notions agree at every point of $A$ , including relative endpoints and isolated points, and hence the on-set notions agree; on the empty set both are vacuous.

step 1.1∎

Depends on

Used by

A lower semicontinuous function on [0,1] with no maximum Counterexample

Dependency tree · two levels

30 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)