Theorems · Theorem · real analysis

Monotone.continuousAt_iff_leftLim_eq_rightLim

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : ConditionallyCompleteLinearOrder β]
  [inst_2 : TopologicalSpace β] [OrderTopology β] {f : α → β},
  Monotone f →
    ∀ {x : α} [inst_4 : TopologicalSpace α] [OrderTopology α],
      ContinuousAt f x ↔ Function.leftLim f x = Function.rightLim f x

A monotone function is continuous at a point if and only if its left and right limits coincide.

Defined in

Mathlib.Topology.Order.LeftRightLim

Cited by

2 results in Mathlib

Foundations

Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyTopologicalSpaceOrderTopology