Theorems · Theorem · real analysis

LocallyBoundedVariationOn.ae_differentiableWithinAt

∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {f : ℝ → V}
  {s : Set ℝ},
  LocallyBoundedVariationOn f s →
    MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x

A real function into a finite-dimensional real vector space with bounded variation on a set is differentiable almost everywhere in this set.

Defined in

Mathlib.Analysis.BoundedVariation

Cited by

1 results in Mathlib

Foundations

Depth 269 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceFiniteDimensional