Theorems · Theorem · global analysis

ContDiffAt.to_localInverse

∀ {𝕂 : Type u_1} [inst : RCLike 𝕂] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕂 E]
  {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕂 F] [inst_5 : CompleteSpace E] {f : E → F}
  {f' : E ≃L[𝕂] F} {a : E} {n : WithTop ℕ∞} (hf : ContDiffAt 𝕂 n f a) (hf' : HasFDerivAt f (↑f') a) (hn : n ≠ 0),
  ContDiffAt 𝕂 n (hf.localInverse hf' hn) (f a)

Given a ContDiff function over 𝕂 (which is ℝ or ℂ) with an invertible derivative at a, the inverse function (produced by ContDiff.toOpenPartialHomeomorph) is also ContDiff.

Defined in

Mathlib.Analysis.Calculus.InverseFunctionTheorem.ContDiff

Cited by

1 results in Mathlib

Foundations

Depth 204 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

RCLikeNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace