Theorems · Theorem · general topology

Continuous.of_ordContinuous

∀ {X : Type u_1} [inst : ConditionallyCompleteLinearOrder X] [inst_1 : TopologicalSpace X] [OrderTopology X]
  {Y : Type u_2} [inst_3 : ConditionallyCompleteLinearOrder Y] [inst_4 : TopologicalSpace Y] [OrderTopology Y]
  {f : X → Y}, LeftOrdContinuous f → RightOrdContinuous f → Continuous f

A function that is order-theoretically both left- and right-continuous is continuous, assuming the function is between conditionally complete linear orders with order topologies.

Defined in

Mathlib.Topology.Order.Basic

Cited by

0 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

ConditionallyCompleteLinearOrderTopologicalSpaceOrderTopologyConditionallyCompleteLinearOrderTopologicalSpaceOrderTopology