Theorems · Theorem · general topology

continuousWithinAt_right_of_monotoneOn_of_exists_between

∀ {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] {f : α → β} {s : Set α} {a : α},
  MonotoneOn f s →
    s ∈ nhdsWithin a (Set.Ici a) → (∀ b > f a, ∃ c ∈ s, f c ∈ Set.Ioo (f a) b) → ContinuousWithinAt f (Set.Ici a) a

If f is a monotone function on a right neighborhood of a and the image of this neighborhood under f meets every interval (f a, b), b > f a, then f is continuous at a from the right. The assumption hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioo (f a) b cannot be replaced by the weaker assumption hfs : ∀ b > f a, ∃ c ∈ s, f c ∈ Ioc (f a) b we use for strictly monotone functions because otherwise the function ceil : ℝ → ℤ would be a counter-example at a = 0.

Defined in

Mathlib.Topology.Order.MonotoneContinuity

Cited by

3 results in Mathlib

Foundations

Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderTopologicalSpaceOrderTopology