Theorems · Theorem · general topology

continuousOn_Icc_extendFrom_Ioo

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [DenselyOrdered α]
  [OrderTopology α] [inst_4 : TopologicalSpace β] {f : α → β} {a b : α} {la lb : β} [RegularSpace β],
  ContinuousOn f (Set.Ioo a b) →
    Filter.Tendsto f (nhdsWithin a (Set.Ioi a)) (nhds la) →
      Filter.Tendsto f (nhdsWithin b (Set.Iio b)) (nhds lb) → ContinuousOn (extendFrom (Set.Ioo a b) f) (Set.Icc a b)

Defined in

Mathlib.Topology.Order.ExtendFrom

Cited by

2 results in Mathlib

Foundations

Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceLinearOrderDenselyOrderedOrderTopologyTopologicalSpaceRegularSpace