Theorems · Theorem · real analysis

ContDiffWithinAt.isSymmSndFDerivWithinAt

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} {F : Type u_3} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F} {x : E}
  {n : WithTop ℕ∞},
  ContDiffWithinAt 𝕜 n f s x →
    minSmoothness 𝕜 2 ≤ n → UniqueDiffOn 𝕜 s → x ∈ closure (interior s) → x ∈ s → IsSymmSndFDerivWithinAt 𝕜 f s x

If a function is C^2 within a set at a point, and accumulated by points in the interior of the set, then its second derivative there is symmetric. Over a field different from ℝ or ℂ, we should require that the function is analytic.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Symmetric

Cited by

6 results in Mathlib

Foundations

Depth 210 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace