continuousOn_deriv_of_isOpen — Mathlib

English

If h and its derivative satisfy ContDiffOn, then the derivative's derivative within domain is ContDiffOn.

Русский

Если h и его производная удовлетворяют ContDiffOn, то производная внутри области также ContDiffOn.

LaTeX

$$$ContDiffOn\ 𝕜\ n\ (derivWithin\ 𝕜\ f\ s)\ s \to\ ContDiffOn\ 𝕜\ (n-1) (derivWithin f\ s)$$$

Lean4

theorem continuousOn_deriv_of_isOpen (h : ContDiffOn 𝕜 n f₂ s₂) (hs : IsOpen s₂) (hn : 1 ≤ n) :
    ContinuousOn (deriv f₂) s₂ := by
  rw [show (1 : WithTop ℕ∞) = 0 + 1 from rfl] at hn
  exact ((contDiffOn_succ_iff_deriv_of_isOpen hs).1 (h.of_le hn)).2.2.continuousOn

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