Theorems · Theorem · real analysis

AnalyticOn.contDiffOn

∀ {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} {n : WithTop ℕ∞}, AnalyticOn 𝕜 f s → UniqueDiffOn 𝕜 s → ContDiffOn 𝕜 n f s

On a set with unique differentiability, an analytic function is automatically C^ω, as its successive derivatives are also analytic. This does not require completeness of the space. See also AnalyticOn.contDiffOn_of_completeSpace.

Defined in

Mathlib.Analysis.Calculus.ContDiff.Defs

Cited by

5 results in Mathlib

Foundations

Depth 195 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace