Theorems · Theorem · real analysis
MonotoneOn.ae_differentiableWithinAt_of_mem
∀ {f : ℝ → ℝ} {s : Set ℝ}, MonotoneOn f s → ∀ᵐ (x : ℝ), x ∈ s → DifferentiableWithinAt ℝ f s xA real function which is monotone on a set is differentiable Lebesgue-almost everywhere on
this set. This version does not assume that s is measurable. For a formulation with
volume.restrict s assuming that s is measurable, see MonotoneOn.ae_differentiableWithinAt.
- Defined in
- Mathlib.Analysis.Calculus.Monotone
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 265 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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
- MeasureTheory.Measurestatement · cited by 10,939
- Filter.Eventuallystatement · cited by 3,134
- MeasureTheory.aestatement · cited by 2,352
- LT.lt.leproof · cited by 2,189
- nhdsWithinproof · cited by 1,912
- Set.Iccproof · cited by 1,702
- Filter.univ_mem'proof · cited by 1,672
- Filter.mp_memproof · cited by 1,537
- Monotoneproof · cited by 1,397
- MeasureTheory.MeasureSpace.volumestatement · cited by 1,323
Cited by3
Results whose statement or proof uses this declaration.
- BoundedVariationOn.intervalIntegrable_derivproof · cited by 1
- LocallyBoundedVariationOn.ae_differentiableWithinAt_of_mem_realproof · cited by 1
- MonotoneOn.ae_differentiableWithinAtproof · cited by 0