Theorems · Theorem · real analysis
leftLim_leftLim
∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace β] [inst_2 : TopologicalSpace α]
[OrderTopology α] [T3Space β] {f : α → β} {a : α},
Filter.Tendsto f (nhdsWithin a (Set.Iio a)) (nhds (Function.leftLim f a)) →
Function.leftLim (Function.leftLim f) a = Function.leftLim f a- Defined in
- Mathlib.Topology.Order.LeftRightLim
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 94 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- LinearOrderstatement and proof · cited by 8,572
- nhdsstatement and proof · cited by 5,554
- Filter.Tendstostatement and proof · cited by 3,814
- nhdsWithinstatement and proof · cited by 1,912
- OrderTopologystatement and proof · cited by 1,355
- Set.Iiostatement and proof · cited by 1,166
- Function.leftLimstatement and proof · cited by 60
- T3Spacestatement and proof · cited by 51
- ContinuousWithinAt.leftLim_eqproof · cited by 4
- continuousWithinAt_leftLim_Iicproof · cited by 2
Cited by2
Results whose statement or proof uses this declaration.
- BoundedVariationOn.continuousWithinAt_leftLimproof · cited by 1
- rightLim_rightLimproof · cited by 0