Theorems · Theorem · global analysis

AnalyticOn.fderivWithin

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u} [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] {F : Type v} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F}
  {s : Set E}, AnalyticOn 𝕜 f s → UniqueDiffOn 𝕜 s → AnalyticOn 𝕜 (fderivWithin 𝕜 f s) s

If a function is analytic within a set with unique differentials, then so is its derivative. Note that this theorem does not require completeness of the space.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Analytic

Cited by

3 results in Mathlib

Foundations

Depth 191 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace