Theorems · Theorem · general topology

intermediate_value_Iio

∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α]
  [DenselyOrdered α] {δ : Type u_1} [inst_4 : LinearOrder δ] [inst_5 : TopologicalSpace δ] [OrderClosedTopology δ]
  {a : α} {f : α → δ},
  ContinuousOn f (Set.Iic a) → Filter.Tendsto f Filter.atBot Filter.atBot → Set.Iio (f a) ⊆ f '' Set.Iio a

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

1 results in Mathlib

Foundations

Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopology