Theorems · Theorem · real analysis

BoundedVariationOn.exists_tendsto_left_of_filter

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] [CompleteSpace E] {f : α → E}
  {s : Set α},
  BoundedVariationOn f s →
    ∀ (L : Filter α), (∀ y ∈ s, s ∩ Set.Ici y ∈ L) → s.Nonempty → ∃ l, Filter.Tendsto f L (nhds l)

A bounded variation function has a limit on its left within a set. Version with a general filter, covering both left neighborhoods of points and atTop.

Defined in

Mathlib.Topology.EMetricSpace.BoundedVariation

Cited by

2 results in Mathlib

Foundations

Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderPseudoEMetricSpaceCompleteSpace