Theorems · Theorem · real analysis

BoundedVariationOn.tendsto_eVariationOn_Ici_zero_of_filter

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] {f : α → E} {s : Set α},
  BoundedVariationOn f s →
    ∀ (L : Filter α), (∀ y ∈ s, s ∩ Set.Ici y ∈ L) → Filter.Tendsto (fun y => eVariationOn f (s ∩ Set.Ici y)) L (nhds 0)

If a function has bounded variation, then the variation on closed semi-infinite intervals tends to 0. We give a version with a generic filter, that applies both to left-neighborhoods of points and to atTop.

Defined in

Mathlib.Topology.EMetricSpace.BoundedVariation

Cited by

3 results in Mathlib

Foundations

Depth 157 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderPseudoEMetricSpace