Theorems · Theorem · global analysis

DifferentiableOn.union_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} {s t : Set E},
  DifferentiableOn 𝕜 f s → DifferentiableOn 𝕜 f t → IsOpen s → IsOpen t → DifferentiableOn 𝕜 f (s ∪ t)

If a function is differentiable on two open sets, it is also differentiable on their union.

Defined in

Mathlib.Analysis.Calculus.FDeriv.Basic

Cited by

2 results in Mathlib

Foundations

Depth 165 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

NontriviallyNormedFieldAddCommGroupModuleTopologicalSpaceAddCommGroupModuleTopologicalSpace