Theorems · Theorem · global analysis
DifferentiableOn.iUnion_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), DifferentiableOn 𝕜 f (s i)) → (∀ (i : ι), IsOpen (s i)) → DifferentiableOn 𝕜 f (⋃ i, s i)If a function is differentiable on open sets s i, it is 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
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Modulestatement and proof · cited by 20,661
- AddCommGroupstatement and proof · cited by 12,871
- NontriviallyNormedFieldstatement and proof · cited by 8,742
- Set.rangeproof · cited by 4,705
- Set.iUnionstatement and proof · cited by 2,483
- IsOpenstatement and proof · cited by 2,400
- IsOpen.mem_nhdsproof · cited by 470
- DifferentiableOnstatement and proof · cited by 419
- DifferentiableAt.differentiableWithinAtproof · cited by 96
- DifferentiableOn.differentiableAtproof · cited by 26
Cited by2
Results whose statement or proof uses this declaration.
- differentiable_of_differentiableOn_iUnion_of_isOpenproof · cited by 0
- differentiableOn_iUnion_iff_of_isOpenproof · cited by 0