File: Mathlib/Topology/MetricSpace/Equicontinuity.lean

full_name: Metric.equicontinuousAt_iff_right
code: /-- Characterization of equicontinuity for families of functions taking values in a (pseudo) metric
space. -/
theorem equicontinuousAt_iff_right {ι : Type*} [TopologicalSpace β] {F : ι → β → α} {x₀ : β} :
    EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) < ε 
goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∀ᶠ (x : β) in 𝓝 x₀, ∀ (i : ι), dist (F i x₀) (F i x) < ε

full_name: Metric.equicontinuousAt_iff
code: /-- Characterization of equicontinuity for families of functions between (pseudo) metric spaces. -/
theorem equicontinuousAt_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} {x₀ : β} :
    EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ x, dist x x₀ < δ → ∀ i, dist (F i x₀) (F i x) < ε 
goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x : β), dist x x₀ < δ → ∀ (i : ι), dist (F i x₀) (F i x) < ε

full_name: Metric.equicontinuousAt_iff_pair
code: /-- Reformulation of `equicontinuousAt_iff_pair` for families of functions taking values in a
(pseudo) metric space. -/
protected theorem equicontinuousAt_iff_pair {ι : Type*} [TopologicalSpace β] {F : ι → β → α}
    {x₀ : β} :
    EquicontinuousAt F x₀ ↔
      ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ i, dist (F i x) (F i x') < ε 
goal: EquicontinuousAt F x₀ ↔ ∀ ε > 0, ∃ U ∈ 𝓝 x₀, ∀ x ∈ U, ∀ x' ∈ U, ∀ (i : ι), dist (F i x) (F i x') < ε

full_name: Metric.uniformEquicontinuous_iff_right
code: /-- Characterization of uniform equicontinuity for families of functions taking values in a
(pseudo) metric space. -/
theorem uniformEquicontinuous_iff_right {ι : Type*} [UniformSpace β] {F : ι → β → α} :
    UniformEquicontinuous F ↔ ∀ ε > 0, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, dist (F i xy.1) (F i xy.2) < ε 
goal: UniformEquicontinuous F ↔ ∀ ε > 0, ∀ᶠ (xy : β × β) in 𝓤 β, ∀ (i : ι), dist (F i xy.1) (F i xy.2) < ε

full_name: Metric.uniformEquicontinuous_iff
code: /-- Characterization of uniform equicontinuity for families of functions between
(pseudo) metric spaces. -/
theorem uniformEquicontinuous_iff {ι : Type*} [PseudoMetricSpace β] {F : ι → β → α} :
    UniformEquicontinuous F ↔
      ∀ ε > 0, ∃ δ > 0, ∀ x y, dist x y < δ → ∀ i, dist (F i x) (F i y) < ε 
goal: UniformEquicontinuous F ↔ ∀ ε > 0, ∃ δ > 0, ∀ (x y : β), dist x y < δ → ∀ (i : ι), dist (F i x) (F i y) < ε

full_name: Metric.equicontinuousAt_of_continuity_modulus
code: /-- For a family of functions to a (pseudo) metric spaces, a convenient way to prove
equicontinuity at a point is to show that all of the functions share a common *local* continuity
modulus. -/
theorem equicontinuousAt_of_continuity_modulus {ι : Type*} [TopologicalSpace β] {x₀ : β}
    (b : β → ℝ) (b_lim : Tendsto b (𝓝 x₀) (𝓝 0)) (F : ι → β → α)
    (H : ∀ᶠ x in 𝓝 x₀, ∀ i, dist (F i x₀) (F i x) ≤ b x) : EquicontinuousAt F x₀ 
goal: EquicontinuousAt F x₀

full_name: Metric.uniformEquicontinuous_of_continuity_modulus
code: /-- For a family of functions between (pseudo) metric spaces, a convenient way to prove
uniform equicontinuity is to show that all of the functions share a common *global* continuity
modulus. -/
theorem uniformEquicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
    (b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
    (H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : UniformEquicontinuous F 
goal: UniformEquicontinuous F

full_name: Metric.equicontinuous_of_continuity_modulus
code: /-- 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. -/
theorem equicontinuous_of_continuity_modulus {ι : Type*} [PseudoMetricSpace β] (b : ℝ → ℝ)
    (b_lim : Tendsto b (𝓝 0) (𝓝 0)) (F : ι → β → α)
    (H : ∀ (x y : β) (i), dist (F i x) (F i y) ≤ b (dist x y)) : Equicontinuous F 
goal: Equicontinuous F