Theorems · Theorem · real analysis

ContDiffOn.iUnion_of_isOpen

∀ {𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} [inst : NontriviallyNormedField 𝕜] [inst_1 : NormedAddCommGroup E]
  [inst_2 : NormedSpace 𝕜 E] [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F} {n : WithTop ℕ∞}
  {ι : Type u_5} {s : ι → Set E},
  (∀ (i : ι), ContDiffOn 𝕜 n f (s i)) → (∀ (i : ι), IsOpen (s i)) → ContDiffOn 𝕜 n f (⋃ i, s i)

If a function is C^k on open sets s i, it is C^k on their union

Defined in

Mathlib.Analysis.Calculus.ContDiff.Basic

Cited by

2 results in Mathlib

Foundations

Depth 180 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldNormedAddCommGroupNormedSpaceNormedAddCommGroupNormedSpace