Theorems · Theorem · general topology

ContinuousOn.surjOn_Icc

∀ {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]
  {s : Set α} [hs : s.OrdConnected] {f : α → δ},
  ContinuousOn f s → ∀ {a b : α}, a ∈ s → b ∈ s → Set.SurjOn f s (Set.Icc (f a) (f b))

Intermediate value theorem: if f is continuous on an order-connected set s and a, b are two points of this set, then f sends s to a superset of Icc (f a) (f b).

Defined in

Mathlib.Topology.Order.IntermediateValue

Cited by

1 results in Mathlib

Foundations

Depth 88 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceConditionallyCompleteLinearOrderOrderTopologyDenselyOrderedLinearOrderTopologicalSpaceOrderClosedTopologySet.OrdConnected