Theorems · Theorem · general topology

Metric.equicontinuous_of_continuity_modulus

∀ {α : Type u_1} {β : Type u_2} [inst : PseudoMetricSpace α] {ι : Type u_4} [inst_1 : PseudoMetricSpace β] (b : ℝ → ℝ),
  Filter.Tendsto b (nhds 0) (nhds 0) →
    ∀ (F : ι → β → α), (∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)) → Equicontinuous F

For a family of functions between (pseudo) metric spaces, a convenient way to prove equicontinuity is to show that all of the functions share a common global continuity modulus.

Defined in

Mathlib.Topology.MetricSpace.Equicontinuity

Cited by

0 results in Mathlib

Foundations

Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

PseudoMetricSpacePseudoMetricSpace