Theorems · Theorem · real analysis
leftLim_eq_of_tendsto
∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace β] [hα : TopologicalSpace α]
[h'α : OrderTopology α] [T2Space β] {f : α → β} {a : α} {y : β} [h : (nhdsWithin a (Set.Iio a)).NeBot],
Filter.Tendsto f (nhdsWithin a (Set.Iio a)) (nhds y) → Function.leftLim f a = y- Defined in
- Mathlib.Topology.Order.LeftRightLim
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 92 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- T2Spacestatement and proof · cited by 1,351
- Set.Iiostatement and proof · cited by 1,166
- Filter.NeBotstatement and proof · cited by 853
- Function.leftLimstatement · cited by 60
- Filter.limUnderproof · cited by 47
- Filter.neBot_iffproof · cited by 27
Cited by5
Results whose statement or proof uses this declaration.
- ContinuousWithinAt.leftLim_eqproof · cited by 4
- leftLim_rightLimproof · cited by 2
- rightLim_eq_of_tendstoproof · cited by 2
- Monotone.leftLim_eq_sSupproof · cited by 2
- variationOnFromTo.leftLim_eqproof · cited by 0