Theorems · Theorem · general topology

continuousWithinAt_left_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.Iic a) → (∀ b < f a, ∃ c ∈ s, f c ∈ Set.Ioo b (f a)) → ContinuousWithinAt f (Set.Iic a) a

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

Defined in

Mathlib.Topology.Order.MonotoneContinuity

Cited by

1 results in Mathlib

Foundations

Depth 76 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderTopologicalSpaceOrderTopology