ae_differentiableAt — Mathlib

English

A fiberwise Theta property holds: if f =Θ[l×ˢ l'] g, then for almost every y in l' the function x ↦ f(x,y) is Θ_l with g(x,y).

Русский

Свойство Θ по срезам слева: если f =Θ[l×ˢ l'] g, то для почти всех y в l' срезы удовлетворяют Θ по x.

LaTeX

$$$\forall f,g:\; f =Θ[\mathrm{Filter.instSProd.sprod}(l,l')] g \Rightarrow \forall^\infty y\in l',\; f(\cdot,y) =Θ_l (g(\cdot,y))$$$

Lean4

/-- A real function into a finite-dimensional real vector space with bounded variation
is differentiable almost everywhere. -/
theorem ae_differentiableAt {f : ℝ → V} (h : LocallyBoundedVariationOn f univ) : ∀ᵐ x, DifferentiableAt ℝ f x :=
  by
  filter_upwards [h.ae_differentiableWithinAt_of_mem] with x hx
  rw [differentiableWithinAt_univ] at hx
  exact hx (mem_univ _)

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