LocallyLipschitz — Mathlib

English

For sets s,t and points x∈s, y∈t, diam(s ∪ t) is bounded by diam(s) + edist(x,y) + diam(t).

Русский

Для множеств s, t и точек x ∈ s, y ∈ t диаметр объединения diam(s ∪ t) ограничен diam(s) + edist(x,y) + diam(t).

LaTeX

$$\operatorname{diam}(s \cup t) \le \operatorname{diam}(s) + \operatorname{edist}(x,y) + \operatorname{diam}(t) \, (x \in s, \ y \in t)$$

Lean4

/-- `f : α → β` is called **locally Lipschitz continuous** iff every point `x`
has a neighbourhood on which `f` is Lipschitz. -/
def LocallyLipschitz (f : α → β) : Prop :=
  ∀ x, ∃ K, ∃ t ∈ 𝓝 x, LipschitzOnWith K f t

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