Theorems · Theorem · general topology
ContinuousOn.domRestrict
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set α},
ContinuousOn f s → Continuous (s.domRestrict f)Alias of the forward direction of continuousOn_iff_continuous_domRestrict.
- Defined in
- Mathlib.Topology.ContinuousOn
- Cited by
- 99 results in Mathlib
- Foundations
- Depth 76 from the axioms, rests on 876 definitions · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites7
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
- Set.Elemstatement · cited by 7,166
- Continuousstatement · cited by 2,592
- ContinuousOnstatement · cited by 1,411
- Set.domRestrictstatement · cited by 383
- continuousOn_iff_continuous_domRestrictproof · cited by 51
Cited by99
Results whose statement or proof uses this declaration.
- cfc_applystatement and proof · cited by 40
- cfcₙ_applystatement and proof · cited by 32
- cfcₙ_congrproof · cited by 25
- cfc_congrproof · cited by 20
- cfcₙ_eq_cfcproof · cited by 13
- cfc_constproof · cited by 12
- cfc_casesstatement and proof · cited by 12
- cfc_map_spectrumproof · cited by 11
- cfcₙ_casesstatement and proof · cited by 10
- cfcₙ_mulproof · cited by 10
- cfc_powproof · cited by 8
- cfc_algebraMapproof · cited by 8