Theorems · Theorem · real analysis

contDiffOn_succ_iff_fderiv_apply

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {D : Type uD} [inst_1 : NormedAddCommGroup D]
  [inst_2 : NormedSpace 𝕜 D] {E : Type uE} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {n : WithTop ℕ∞}
  {f : D → E} {s : Set D} [CompleteSpace 𝕜] [FiniteDimensional 𝕜 D],
  UniqueDiffOn 𝕜 s →
    (ContDiffOn 𝕜 (n + 1) f s ↔
      DifferentiableOn 𝕜 f s ∧
        (n = ⊤ → AnalyticOn 𝕜 f s) ∧ ∀ (y : D), ContDiffOn 𝕜 n (fun x => (fderivWithin 𝕜 f s x) y) s)

Defined in

Mathlib.Analysis.Calculus.ContDiff.FiniteDimension

Cited by

0 results in Mathlib

Foundations

Depth 202 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpaceFiniteDimensional