Theorems · Theorem · real analysis

BoundedVariationOn.exists_tendsto_left

∀ {α : Type u_1} [inst : LinearOrder α] {E : Type u_2} [inst_1 : PseudoEMetricSpace E] [CompleteSpace E]
  [inst_3 : TopologicalSpace α] [OrderTopology α] {f : α → E} {s : Set α},
  BoundedVariationOn f s → ∀ (x : α), ∃ l, Filter.Tendsto f (nhdsWithin x (s ∩ Set.Iio x)) (nhds l)

A bounded variation function has a limit on its left within a set.

Defined in

Mathlib.Topology.EMetricSpace.BoundedVariation

Cited by

2 results in Mathlib

Foundations

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

Assumes

LinearOrderPseudoEMetricSpaceCompleteSpaceTopologicalSpaceOrderTopology