Theorems · Theorem · real analysis

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

A function is C^(n + 1) on a domain iff locally, it has a derivative which is C^n (and moreover the function is analytic when n = ω).

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

5 results in Mathlib

Foundations

Depth 188 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace