Theorems · Theorem · real analysis

AnalyticAt.contDiffAt

∀ {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] {f : E → F}
  {x : E} {n : WithTop ℕ∞} [CompleteSpace F], AnalyticAt 𝕜 f x → ContDiffAt 𝕜 n f x

In a complete space, a function which is analytic 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

8 results in Mathlib

Foundations

Depth 197 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpaceCompleteSpace