Theorems · Theorem · real analysis

contDiffOn_succ_iff_hasFDerivWithinAt

∀ {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 : WithTop ℕ∞},
  n ≠ ↑⊤ →
    (ContDiffOn 𝕜 (n + 1) f s ↔
      ∀ x ∈ s,
        ∃ u ∈ nhdsWithin x (insert x s),
          (n = ⊤ → AnalyticOn 𝕜 f u) ∧ ∃ f', (∀ x ∈ u, HasFDerivWithinAt f (f' x) u x) ∧ ContDiffOn 𝕜 n f' u)

A function is C^(n + 1) on a domain iff locally, it has a derivative which is C^n.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

0 results in Mathlib

Foundations

Depth 189 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace