continuousWithinAt — Mathlib

English

If a family is equicontinuous at x0 and f is in the family, then f is continuous at x0.

Русский

Если семейство экконтинуально в x0 и f принадлежит этому семейству, то f непрервен в x0.

LaTeX

$$$\text{Set.EquicontinuousAt.continuousAt_of_mem}$ : "protected theorem continuousAt_of_mem"$$

Lean4

/-- Each function of a family equicontinuous at `x₀` within `S` is continuous at `x₀` within `S`. -/
theorem continuousWithinAt {F : ι → X → α} {S : Set X} {x₀ : X} (h : EquicontinuousWithinAt F S x₀) (i : ι) :
    ContinuousWithinAt (F i) S x₀ :=
  (UniformSpace.hasBasis_nhds _).tendsto_right_iff.2 fun U ⟨hU, _⟩ ↦ (h U hU).mono fun _x hx ↦ hx i

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