ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 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.
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Characteristic functions of open and closed sets are one-sided semicontinuous

Example

Let $n \geq 1$ , let $X \subseteq R^{n}$ , and let $E \subseteq X$ . The characteristic function $1_{E} : X \to R$ is lower semicontinuous when $E$ is relatively open, and upper semicontinuous when $E$ is relatively closed. On $[0, 1]$ , the characteristic function of ${0}$ is upper semicontinuous but discontinuous at zero, and its negative is lower semicontinuous but discontinuous there.

Facts & Assumptions

Given: The relative Euclidean topology and the semicontinuity convention Upper and lower semicontinuity on subsets of $R^{n}$ .

[L1]

Upper semicontinuity is equivalent to relative openness of every strict sublevel set, and lower semicontinuity is equivalent to relative openness of every strict superlevel set (Semicontinuity on $R^{n}$ is characterized by strict open level sets and weak closed level sets).

[L2]

A set is closed exactly when its complement is open (The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement).

Verification

technique · direct

1.1

Strict superlevels of $1_{E}$ are empty, $E$ , or all of $X$ , while strict sublevels are empty, $X ∖ E$ , or all of $X$ . By [L1] and [L2], $1_{E}$ is lower semicontinuous when $E$ is open and upper semicontinuous when $E$ is closed.

L1L2algebra

2.1

For $E = {0}$ in $[0, 1]$ , step 1.1 gives upper semicontinuity, but values at positive points tend to zero rather than the value one at zero. Negation exchanges upper and lower semicontinuity, so $- 1_{E}$ supplies the lower-semicontinuous discontinuous example.

step 1.1algebra∎

Depends on

Used by

FALSE: semicontinuity implies continuity on a compact set False statement

Dependency tree · two levels

8 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)