continuous_deriv — Mathlib

English

Under suitable bounds, derivative of a C^n function within a domain is C^n in both variables.

Русский

При разумных ограничениях производная внутри области гладка по обеим переменным.

LaTeX

$$$ContDiffOn\ 𝕜\ n\ (f)\ s \to\ ContDiffOn\ 𝕜\ n\ ("derivWithin f s")\ s$$$

Lean4

theorem continuous_deriv (h : ContDiff 𝕜 n f₂) (hn : 1 ≤ n) : Continuous (deriv f₂) :=
  by
  rw [show (1 : WithTop ℕ∞) = 0 + 1 from rfl] at hn
  exact (contDiff_succ_iff_deriv.mp (h.of_le hn)).2.2.continuous

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