contDiffWithinAt_compl_self — Mathlib

English

If f is ContDiffOn on n on s and s is open, then fderiv 𝕜 f is ContDiffOn on s for m with m+1 ≤ n.

Русский

Если f ∈ ContDiffOn на s и s открыто, тогда fderiv 𝕜 f ∈ ContDiffOn на s для всех m+1 ≤ n.

LaTeX

$$$\\text{ContDiffOn 𝕜 n f s} \\land \\text{IsOpen } s \\land m + 1 \\le n \\Rightarrow \\text{ContDiffOn 𝕜 m } (fderiv 𝕜 f) s$$$

Lean4

@[simp]
theorem contDiffWithinAt_compl_self : ContDiffWithinAt 𝕜 n f { x }ᶜ x ↔ ContDiffAt 𝕜 n f x := by
  rw [compl_eq_univ_diff, contDiffWithinAt_diff_singleton, contDiffWithinAt_univ]

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