Theorems · Theorem · general topology
continuousOn_of_forall_continuousAt
∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set α},
(∀ x ∈ s, ContinuousAt f x) → ContinuousOn f s- Defined in
- Mathlib.Topology.ContinuousOn
- Cited by
- 30 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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
- ContinuousOnstatement · cited by 1,411
- ContinuousAtstatement and proof · cited by 697
- ContinuousAt.continuousWithinAtproof · cited by 102
Cited by30
Results whose statement or proof uses this declaration.
- intervalIntegral.intervalIntegrable_cpow'proof · cited by 7
- Orientation.oangle_sign_smul_add_rightproof · cited by 6
- EReal.continuous_toENNRealproof · cited by 4
- Complex.GammaIntegral_convergentproof · cited by 3
- intervalIntegral.intervalIntegrable_cpowproof · cited by 2
- integral_gaussian_complexproof · cited by 2
- AffineSubspace.SSameSide.oangle_sign_eqproof · cited by 2
- ZetaAsymptotics.term_welldefproof · cited by 2
- Collinear.oangle_sign_of_sameRay_vsubproof · cited by 2
- mellin_convergent_of_isBigO_scalarproof · cited by 2
- mellin_hasDerivAt_of_isBigO_rpowproof · cited by 2