Theorems · Theorem · real analysis

BoundedVariationOn.continuousWithinAt_variationOnFromTo_Ici

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] {f : α → E}
  [inst_2 : TopologicalSpace α] [OrderTopology α],
  BoundedVariationOn f Set.univ →
    ∀ {a x : α}, ContinuousWithinAt f (Set.Ici x) x → ContinuousWithinAt (variationOnFromTo f Set.univ a) (Set.Ici x) x

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

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Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Realstatement and proof · cited by 25,697
TopologicalSpacestatement and proof · cited by 24,529
ENNRealproof · cited by 9,879
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
nhdsWithinproof · cited by 1,912
Set.Iccproof · cited by 1,702
PseudoEMetricSpacestatement and proof · cited by 1,536
OrderTopologystatement and proof · cited by 1,355
Set.Icistatement and proof · cited by 1,070

Cited by1

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BoundedVariationOn.continuousWithinAt_variationOnFromTo_rightLim_Iciproof · cited by 0