Theorems · Theorem · real analysis

contDiffOn_iff_continuousOn_differentiableOn

∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E}
  {f : E → F} {n : ℕ∞},
  UniqueDiffOn 𝕜 s →
    (ContDiffOn 𝕜 (↑n) f s ↔
      (∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => iteratedFDerivWithin 𝕜 m f s x) s) ∧
        ∀ (m : ℕ), ↑m < n → DifferentiableOn 𝕜 (fun x => iteratedFDerivWithin 𝕜 m f s x) s)

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

3 results in Mathlib

Foundations

Depth 186 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace