Theorems · Theorem · general topology

ContinuousOn.image_Ioi_of_strictMonoOn

∀ {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.Ici a) →
    StrictMonoOn f (Set.Ici a) → Filter.Tendsto f Filter.atTop Filter.atTop → f '' Set.Ioi a = Set.Ioi (f a)

Defined in: Mathlib.Topology.Order.IntermediateValue
Cited by: 3 results in Mathlib
Foundations: Depth 91 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: TopologicalSpace ConditionallyCompleteLinearOrder OrderTopology DenselyOrdered LinearOrder TopologicalSpace OrderClosedTopology

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Cites17

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Setstatement · cited by 53,352
TopologicalSpacestatement and proof · cited by 24,529
LinearOrderstatement and proof · cited by 8,572
Set.imagestatement · cited by 5,609
Filter.Tendstostatement and proof · cited by 3,814
Filter.atTopstatement and proof · cited by 2,405
Set.Ioistatement · cited by 1,463
ContinuousOnstatement and proof · cited by 1,411
OrderTopologystatement and proof · cited by 1,355
Set.Icistatement and proof · cited by 1,070
ConditionallyCompleteLinearOrderstatement and proof · cited by 542
DenselyOrderedstatement and proof · cited by 471

Cited by3

Results whose statement or proof uses this declaration.

Real.image_exp_Ioiproof · cited by 2
Real.image_log_Ioiproof · cited by 0
MeasureTheory.integral_comp_rpow_Ioi_of_pos'proof · cited by 0