Theorems · Theorem · general topology

ContinuousWithinAt.tendsto

∀ {α : Type u_1} {β : Type u_2} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set α}
  {x : α}, ContinuousWithinAt f s x → Filter.Tendsto f (nhdsWithin x s) (nhds (f x))

If a function is continuous within s at x, then it tends to f x within s by definition. We register this fact for use with the dot notation, especially to use Filter.Tendsto.comp as ContinuousWithinAt.comp will have a different meaning.

Defined in

Mathlib.Topology.ContinuousOn

Cited by

44 results in Mathlib

Foundations

Depth 50 from the axioms · uses propext, Quot.sound

Assumes

TopologicalSpaceTopologicalSpace