differentiable_of_differentiableOn_union_of_isOpen

English

Differentiability on a union of open sets equals differentiability on each set component.

Русский

Сумма дифференцируемости по iUnion равна дифференцируемости на каждом множестве-элементе.

LaTeX

$$DifferentiableOn 𝕜 f (⋃ i, s i) ↔ ∀ i, DifferentiableOn 𝕜 f (s i)$$

Lean4

theorem differentiable_of_differentiableOn_union_of_isOpen (hf : DifferentiableOn 𝕜 f s) (hf' : DifferentiableOn 𝕜 f t)
    (hst : s ∪ t = univ) (hs : IsOpen s) (ht : IsOpen t) : Differentiable 𝕜 f :=
  by
  rw [← differentiableOn_univ, ← hst]
  exact hf.union_of_isOpen hf' hs ht

Похожие утверждения