Theorems · Theorem · general topology

StrictMonoOn.continuousWithinAt_left_of_surjOn

∀ {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},
  StrictMonoOn f s → s ∈ nhdsWithin a (Set.Iic a) → Set.SurjOn f s (Set.Iio (f a)) → ContinuousWithinAt f (Set.Iic a) a

If a function f is strictly monotone on a left neighborhood of a and the image of this neighborhood under f includes Iio (f a), then f is continuous at a from the left.

Defined in

Mathlib.Topology.Order.MonotoneContinuity

Cited by

0 results in Mathlib

Foundations

Depth 77 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

LinearOrderTopologicalSpaceOrderTopologyLinearOrderTopologicalSpaceOrderTopology