Theorems · Theorem · real analysis

BoundedVariationOn.continuousWithinAt_leftLim

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] [inst_2 : TopologicalSpace α]
  [OrderTopology α] [CompleteSpace E] [T3Space E] {f : α → E},
  BoundedVariationOn f Set.univ → ∀ {x : α}, ContinuousWithinAt (Function.leftLim f) (Set.Iic x) x

Defined in: Mathlib.Topology.EMetricSpace.BoundedVariation
Cited by: 1 results in Mathlib
Foundations: Depth 161 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: LinearOrder PseudoEMetricSpace TopologicalSpace OrderTopology CompleteSpace T3Space

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Cites19

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
nhdsproof · cited by 5,554
Set.univstatement and proof · cited by 3,945
Filter.Tendstoproof · cited by 3,814
CompleteSpacestatement and proof · cited by 2,532
nhdsWithinproof · cited by 1,912
PseudoEMetricSpacestatement and proof · cited by 1,536
OrderTopologystatement and proof · cited by 1,355
Set.Iioproof · cited by 1,166
Set.Iicstatement · cited by 1,111
ContinuousWithinAtstatement · cited by 512

Cited by1

Results whose statement or proof uses this declaration.

BoundedVariationOn.continuousWithinAt_rightLimproof · cited by 1