Theorems · Theorem · real analysis

ContDiffAt.isSymmSndFDerivAt_of_omega

∀ {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] {f : E → F} {x : E},
  ContDiffAt 𝕜 ⊤ f x → IsSymmSndFDerivAt 𝕜 f x

If a function is analytic at a point, then its second derivative is symmetric.

Defined in: Mathlib.Analysis.Calculus.FDeriv.Symmetric
Cited by: 1 results in Mathlib
Foundations: Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: NontriviallyNormedField NormedAddCommGroup NormedSpace NormedAddCommGroup NormedSpace

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites11

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

NormedAddCommGroupstatement and proof · cited by 15,752
NormedSpacestatement and proof · cited by 12,499
Top.topstatement and proof · cited by 9,680
NontriviallyNormedFieldstatement and proof · cited by 8,742
ENatstatement · cited by 4,985
WithTopstatement · cited by 3,754
Set.mem_univproof · cited by 416
ContDiffAtstatement and proof · cited by 262
uniqueDiffOn_univproof · cited by 66
IsSymmSndFDerivAtstatement · cited by 10
ContDiffWithinAt.isSymmSndFDerivWithinAt_of_omegaproof · cited by 1

Cited by1

Results whose statement or proof uses this declaration.

ContDiffAt.isSymmSndFDerivAtproof · cited by 4