Theorems · Theorem · real analysis

LipschitzOnWith.ae_differentiableWithinAt_of_mem

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  {F : Type u_2} [inst_4 : NormedAddCommGroup F] [inst_5 : NormedSpace ℝ F] {C : NNReal} {s : Set E}
  {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [FiniteDimensional ℝ F] [μ.IsAddHaarMeasure] {f : E → F},
  LipschitzOnWith C f s → ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x

Rademacher's theorem: a function between finite-dimensional real vector spaces which is Lipschitz on a set is differentiable almost everywhere in this set.

Defined in

Mathlib.Analysis.Calculus.Rademacher

Cited by

2 results in Mathlib

Foundations

Depth 309 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceNormedAddCommGroupNormedSpaceFiniteDimensionalFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasure