Theorems · Theorem · real analysis
LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn
∀ {α : Type u_1} [inst : LinearOrder α] {f : α → ℝ} {s : Set α},
LocallyBoundedVariationOn f s → ∃ p q, MonotoneOn p s ∧ MonotoneOn q s ∧ f = p - qIf a real-valued function has bounded variation on a set, then it is a difference of monotone functions there.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 160 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- LinearOrder
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- Realstatement and proof · cited by 25,697
- LinearOrderstatement and proof · cited by 8,572
- MonotoneOnstatement and proof · cited by 311
- LocallyBoundedVariationOnstatement and proof · cited by 41
- variationOnFromToproof · cited by 33
- LocallyBoundedVariationOn.exists_monotoneOn_sub_monotoneOn'proof · cited by 1
Cited by2
Results whose statement or proof uses this declaration.
- BoundedVariationOn.intervalIntegrable_derivproof · cited by 1
- LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_realproof · cited by 1