continuousOn — Mathlib

English

The indexed product of spaces with ultra-uniformities is ultra-uniform.

Русский

Индексированное произведение пространств с ультра-равномерностями сохраняет ультра-равномерность.

LaTeX

$$$IsUltraUniformity(\prod i X_i)$$$

Lean4

/-- A uniform limit on a set of functions which are continuous on this set is itself continuous
on this set. -/
protected theorem continuousOn (h : TendstoUniformlyOn F f p s) (hc : ∀ᶠ n in p, ContinuousOn (F n) s) [NeBot p] :
    ContinuousOn f s :=
  h.tendstoLocallyUniformlyOn.continuousOn hc

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