CounterexampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-21

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

Statement refuted

Every lower semicontinuous real-valued function on a nonempty compact set attains both a minimum and a maximum.

Facts & Assumptions

Given: Define $g : [0, 1] \to R$ by $g (0) = - 1$ and $g (x) = - x$ for $0 < x \leq 1$ , with the one-dimensional convention identified by Upper and lower semicontinuity on subsets of $R^{n}$ , Euclidean semicontinuity agrees with the published real-line definition.

[L1]

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

[L2]

Every lower semicontinuous real-valued function on a nonempty compact Euclidean set is bounded below and attains a minimum (Semicontinuous extreme value theorem on compact Euclidean sets).

Counterexample

technique · direct

1.1

The weak sublevel ${g \leq α}$ is empty for $α < - 1$ , equals ${0, 1}$ at $α = - 1$ , equals ${0} \cup [- α, 1]$ for $- 1 < α < 0$ , and is all of $[0, 1]$ for $α \geq 0$ . Each is relatively closed, so [L1] makes $g$ lower semicontinuous.

L1givenalgebra

2.1

Every value of $g$ is negative, but $g (1 / N) = - 1 / N$ approaches zero through positive natural $N$ . Hence $\sup g ([0, 1]) = 0$ and the supremum is not attained, so $g$ has no maximum.

step 1.1algebra

3.1

The function does attain its minimum $- 1$ at zero, as [L2] requires. It is only the unsupported opposite extremum that fails.

step 2.1L2∎

Depends on

Used by

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Dependency tree · two levels

19 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 5 (standard reference, not scraped)

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