Theorems · Theorem · real analysis

MonotoneOn.ae_differentiableWithinAt

∀ {f : ℝ → ℝ} {s : Set ℝ},
  MonotoneOn f s → MeasurableSet s → ∀ᵐ (x : ℝ) ∂MeasureTheory.volume.restrict s, DifferentiableWithinAt ℝ f s x

A real function which is monotone on a set is differentiable Lebesgue-almost everywhere on this set. This version assumes that s is measurable and uses volume.restrict s. For a formulation without measurability assumption, see MonotoneOn.ae_differentiableWithinAt_of_mem.

Defined in

Mathlib.Analysis.Calculus.Monotone

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Foundations

Depth 266 from the axioms · uses propext, Classical.choice, Quot.sound