Theorems · Theorem · real analysis

BoundedVariationOn.tendsto_eVariationOn_Icc_zero_left

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] [inst_2 : TopologicalSpace α]
  [OrderTopology α] {f : α → E} {s : Set α},
  BoundedVariationOn f s →
    ∀ {x : α},
      ContinuousWithinAt f (s ∩ Set.Iic x) x →
        Filter.Tendsto (fun y => eVariationOn f (s ∩ Set.Icc y x)) (nhdsWithin x s) (nhds 0)

If a function has bounded variation and is left-continuous at a point, then the variation on small closed intervals to the left of this point tends to 0.

Defined in

Mathlib.Topology.EMetricSpace.BoundedVariation

Cited by

1 results in Mathlib

Foundations

Depth 159 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderPseudoEMetricSpaceTopologicalSpaceOrderTopology