continuousOn — Mathlib

English

If a family F is equicontinuous on a set S, then for each i the restricted function F_i|_S is continuous on S.

Русский

Если семейство функций F строго равноконтинуально на области S, то каждая F_i ограничена на S и непрерывна на S.

LaTeX

$$$\\forall i : ι,\\ F_i|_S \\in C(S, α)$$$

Lean4

/-- Each function of a family equicontinuous on `S` is continuous on `S`. -/
theorem continuousOn {F : ι → X → α} {S : Set X} (h : EquicontinuousOn F S) (i : ι) : ContinuousOn (F i) S := fun x hx ↦
  (h x hx).continuousWithinAt i

Похожие утверждения