ae_differentiableWithinAt — Mathlib

English

If f is Theta relative to g, then f composed with fst is Theta relative to g composed with fst.

Русский

Если f =Θ[l] g, то f ∘ fst = Θ[l×ˢ] g ∘ fst.

LaTeX

$$$\forall f,g:\; f =Θ[l] g \Rightarrow (f \circ Prod.fst) =Θ[l' \timesˢ l] (g \circ Prod.fst)$$$

Lean4

/-- A real function into a finite-dimensional real vector space with bounded variation on a set
is differentiable almost everywhere in this set. -/
theorem ae_differentiableWithinAt {f : ℝ → V} {s : Set ℝ} (h : LocallyBoundedVariationOn f s) (hs : MeasurableSet s) :
    ∀ᵐ x ∂volume.restrict s, DifferentiableWithinAt ℝ f s x :=
  by
  rw [ae_restrict_iff' hs]
  exact h.ae_differentiableWithinAt_of_mem

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