Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.eq_iteratedFDerivWithin_of_uniqueDiffOn

∀ {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∞} {p : E → FormalMultilinearSeries 𝕜 E F},
  HasFTaylorSeriesUpToOn n f p s → ∀ {m : ℕ}, ↑m ≤ n → UniqueDiffOn 𝕜 s → x ∈ s → p x m = iteratedFDerivWithin 𝕜 m f s x

On a set with unique differentiability, any choice of iterated differential has to coincide with the one we have chosen in iteratedFDerivWithin 𝕜 m f s.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Cited by

8 results in Mathlib

Foundations

Depth 179 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace