Theorems · Theorem · real analysis

LipschitzOnWith.ae_differentiableWithinAt_real

∀ {V : Type u_1} [inst : NormedAddCommGroup V] [inst_1 : NormedSpace ℝ V] [FiniteDimensional ℝ V] {C : NNReal}
  {f : ℝ → V} {s : Set ℝ},
  LipschitzOnWith C 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 which is Lipschitz on a set is differentiable almost everywhere in this set. For the general Rademacher theorem assuming that the source space is finite dimensional, see LipschitzOnWith.ae_differentiableWithinAt.

Defined in

Mathlib.Analysis.BoundedVariation

Cited by

0 results in Mathlib

Foundations

Depth 270 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceFiniteDimensional