Theorems · Theorem · real analysis

AnalyticWithinAt.contDiffWithinAt

∀ {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∞} [CompleteSpace F], AnalyticWithinAt 𝕜 f s x → ContDiffWithinAt 𝕜 n f s x

In a complete space, a function which is analytic within a set at a point is also C^ω there. Note that the same statement for AnalyticOn does not require completeness, see AnalyticOn.contDiffOn.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

2 results in Mathlib

Foundations

Depth 196 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace