TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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Semicontinuity on $R^{n}$ is characterized by strict open level sets and weak closed level sets

Statement

Let $n \geq 1$ , let $A \subseteq R^{n}$ , and let $f : A \to R$ .

Upper semicontinuity is equivalent to relative openness of every strict sublevel set ${f < α}$ and to relative closedness of every weak superlevel set ${f \geq α}$ .
Lower semicontinuity is equivalent to relative openness of every strict superlevel set and to relative closedness of every weak sublevel set.

Relative openness and closedness refer to the subspace topology on $A$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace, 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).

Facts & Assumptions

Given: The subset and function in the Statement, with Euclidean balls as in Open ball, closed ball and sphere in a metric space.

[F1]

The function $f$ 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 lower clause is its one-sided dual (Upper and lower semicontinuity on subsets of $R^{n}$ ).

Proof

technique · direct

1.1

For the upper-semicontinuous forward direction, if $f (c) < α$ , apply [F1] with $ε = α - f (c)$ to obtain a relative ball inside ${f < α}$ . Conversely, if all strict sublevels are relatively open, the sublevel ${f < f (c) + ε}$ supplies the ball required by [F1].

F1algebra

2.1

Taking complements in $A$ converts the relatively open strict sublevels of step 1.1 into relatively closed weak superlevels and conversely. Thus both upper-semicontinuity characterisations hold in both directions.

step 1.1F1

3.1

Apply steps 1.1 and 2.1 to $- f$ . The lower clause for $f$ is the upper clause for $- f$ , strict superlevels of $f$ are strict sublevels of $- f$ , and weak sublevels of $f$ are weak superlevels of $- f$ . This gives both lower-semicontinuity equivalences in both directions.

step 1.1step 2.1F1algebra∎

Depends on

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Sources

D. Bertsekas, MIT 6.253 Convex Analysis and Optimization, Lectures 2 and 4 (standard reference, not scraped)

Semicontinuity on RnRn is characterized by strict open level sets and weak closed level sets