Theorems · Theorem · real analysis

LipschitzOnWith.ae_differentiableWithinAt_of_mem_pi

∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [inst_2 : MeasurableSpace E] [BorelSpace E]
  {C : NNReal} {μ : MeasureTheory.Measure E} [FiniteDimensional ℝ E] [μ.IsAddHaarMeasure] {ι : Type u_3}
  [inst_6 : Fintype ι] {f : E → ι → ℝ} {s : Set E},
  LipschitzOnWith C f s → ∀ᵐ (x : E) ∂μ, x ∈ s → DifferentiableWithinAt ℝ f s x

A function on a finite-dimensional space which is Lipschitz on a set and taking values in a product space is differentiable almost everywhere in this set. Superseded by LipschitzOnWith.ae_differentiableWithinAt_of_mem which works for functions taking value in any finite-dimensional space.

Defined in

Mathlib.Analysis.Calculus.Rademacher

Cited by

1 results in Mathlib

Foundations

Depth 308 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NormedAddCommGroupNormedSpaceMeasurableSpaceBorelSpaceFiniteDimensionalMeasureTheory.Measure.IsAddHaarMeasureFintype