Theorems · Theorem · global analysis

differentiableOn_iUnion_iff_of_isOpen

∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E]
  [inst_3 : TopologicalSpace E] {F : Type u_3} [inst_4 : AddCommGroup F] [inst_5 : Module 𝕜 F]
  [inst_6 : TopologicalSpace F] {f : E → F} {ι : Type u_4} {s : ι → Set E},
  (∀ (i : ι), IsOpen (s i)) → (DifferentiableOn 𝕜 f (⋃ i, s i) ↔ ∀ (i : ι), DifferentiableOn 𝕜 f (s i))

A function is differentiable on a union of open sets s i iff it is differentiable on each s i.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Basic

Cited by

0 results in Mathlib

Foundations

Depth 166 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpace