Theorems · Theorem · real analysis

contDiffWithinAt_iff_of_ne_infty

∀ {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 f s x ↔
      ∃ u ∈ nhdsWithin x (insert x s),
        ∃ p, HasFTaylorSeriesUpToOn n f p u ∧ (n = ⊤ → ∀ (i : ℕ), AnalyticOn 𝕜 (fun x => p x i) u))

When n is either a natural number or ω, one can characterize the property of being C^n as the existence of a neighborhood on which there is a Taylor series up to order n, requiring in addition that its terms are analytic in the ω case.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

1 results in Mathlib

Foundations

Depth 177 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace