Alphabeta MathHow 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.
A function on a subset of is continuous at iff its oscillation there is , and every oscillation superlevel set is closed
Statement
For , is continuous at if and only if . If is bounded, then for every , the relative superlevel set is closed in .
Facts & Assumptions
Given: .
Metric continuity and balls are Continuity of a map between metric spaces, at a point and globally, in the - form and Open ball, closed ball and sphere in a metric space.
Oscillation is Oscillation of a real function on subsets of and at a point.
Proof
If is continuous at , choose a ball on which ; pairwise differences are then below , so the ball oscillation is at most and .
If , choose with ball oscillation below . Holding one point at gives , proving continuity.
If , choose with . Every has a sufficiently small ball contained in , so . Thus the sublevel set is relatively open.
Steps 1.1 and 1.2 give the equivalence; step 1.3 makes the complementary superlevel set closed.
Depends on
- Oscillation of a real function on subsets of $\mathbb{R}^m$ and at a point
- Continuity of a map between metric spaces, at a point and globally, in the $\varepsilon$-$\delta$ form
- Open ball, closed ball and sphere in a metric space
- 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
- Each $\lVert\cdot\rVert_p$ is a norm on $\mathbb{R}^n$, and the induced metrics are exactly $d_1$, $d_2$ and $d_\infty$ of the published metric-spaces page
- Lower bound, bounded below, bounded set
Used by
Dependency tree · two levels
37 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
- J. Lebl, Basic Analysis, Riemann Integral in Several Variables (standard reference, not scraped)
- J. Lebl, Basic Analysis, The Riemann-Lebesgue Criterion (standard reference, not scraped)