DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-11

How 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.

Variation function and positive and negative variations

Definition

Let $f : [a, b] \to R$ have bounded variation (Bounded variation and total variation on an interval). Its variation function is

$V_{f} (x) : = {Var}_{[a, x]} (f) (x \in [a, b]),$

so $V_{f} (a) = 0$ . This is defined on every subinterval by Total variation is additive over adjacent subintervals and decreases under restriction and the singleton convention. The positive variation and negative variation of $f$ are

$P_{f} (x) : = \frac{V_{f} (x) + f (x) - f (a)}{2}, N_{f} (x) : = \frac{V_{f} (x) - f (x) + f (a)}{2} .$

The names refer to upward and downward accumulated variation; their monotonicity and the resulting decomposition are proved separately.

Depends on

Used by

The positive and negative variations are nondecreasing and give the Jordan identities Lemma

Dependency tree · two levels

12 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 (standard reference, not scraped)
William F. Trench, Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)