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.
Bounded-derivative design correction
The proposed counterclaim “everywhere differentiable with bounded derivative need not be absolutely continuous” is false on a compact interval. Under the stated continuity and derivative hypotheses, If is continuous on an interval and at every interior point, then for all , so is Lipschitz with constant and uniformly continuous on gives Lipschitz continuity, and implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation then gives absolute continuity.
Depends on
- If $f$ is continuous on an interval $I$ and $|f'| \le M$ at every interior point, then $|f(x) - f(y)| \le M|x-y|$ for all $x,y \in I$, so $f$ is Lipschitz with constant $M$ and uniformly continuous on $I$
- $C^1$ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
21 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
- Christopher Heil, Absolute Continuity and the Banach--Zaretsky Theorem, §3.3 (standard reference, not scraped)