LocallyLipschitzOn — Mathlib

English

If s and t have nonempty intersection, then diam(s ∪ t) ≤ diam(s) + diam(t).

Русский

Если пересечение s и t непусто, то диаметр объединения diam(s ∪ t) ≤ diam(s) + diam(t).

LaTeX

$$\operatorname{diam}(s \cup t) \le \operatorname{diam}(s) + \operatorname{diam}(t) \text{ if } (s \cap t) \neq \emptyset$$

Lean4

/-- `f : α → β` is called **locally Lipschitz continuous** on `s` iff every point `x` of `s`
has a neighbourhood within `s` on which `f` is Lipschitz. -/
def LocallyLipschitzOn (s : Set α) (f : α → β) : Prop :=
  ∀ ⦃x⦄, x ∈ s → ∃ K, ∃ t ∈ 𝓝[s] x, LipschitzOnWith K f t

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