Theorems · Theorem · real analysis

HasFTaylorSeriesUpToOn.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∞} {p : E → FormalMultilinearSeries 𝕜 E F},
  HasFTaylorSeriesUpToOn n f p s →
    n ≠ 0 → x ∈ s → HasFDerivWithinAt f ((continuousMultilinearCurryFin1 𝕜 E F) (p x 1)) s x

If a function has a Taylor series at order at least 1, then the term of order 1 of this series is a derivative of f.

Defined in

Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries

Cited by

6 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace