Theorems · Theorem · real analysis

ContDiffOn.ftaylorSeriesWithin

∀ {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∞},
  ContDiffOn 𝕜 n f s → UniqueDiffOn 𝕜 s → HasFTaylorSeriesUpToOn n f (ftaylorSeriesWithin 𝕜 f s) s

When a function is C^n in a set s of unique differentiability, it admits ftaylorSeriesWithin 𝕜 f s as a Taylor series up to order n in s.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

11 results in Mathlib

Foundations

Depth 184 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace