Theorems · Theorem · general topology

continuousWithinAt_left_of_monotoneOn_of_closure_image_mem_nhdsWithin

∀ {α : Type u_1} {β : Type u_2} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
  [inst_3 : LinearOrder β] [inst_4 : TopologicalSpace β] [OrderTopology β] [DenselyOrdered β] {f : α → β} {s : Set α}
  {a : α},
  MonotoneOn f s →
    s ∈ nhdsWithin a (Set.Iic a) →
      closure (f '' s) ∈ nhdsWithin (f a) (Set.Iic (f a)) → ContinuousWithinAt f (Set.Iic a) a

If a function f with a densely ordered codomain is monotone on a left neighborhood of a and the closure of the image of this neighborhood under f is a left neighborhood of f a, then f is continuous at a from the left

Defined in

Mathlib.Topology.Order.MonotoneContinuity

Cited by

2 results in Mathlib

Foundations

Depth 78 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderTopologicalSpaceOrderTopologyDenselyOrdered