div — Mathlib

English

If hf : LipschitzOnWith Kf f s and hg : LipschitzOnWith Kg g s, then the quotient f(x)/g(x) is LipschitzOnWith with constant Kf+Kg on s.

Русский

Если hf и hg Lipшиц OnWith, то f(x)/g(x) липшицево на s с константой Kf+Kg.

LaTeX

$$$\\mathrm{LipschitzOnWith}\\ K_f f\\ s \\land \\mathrm{LipschitzOnWith}\\ K_g g\\ s \\Rightarrow \\mathrm{LipschitzOnWith}\\ (K_f+K_g)\\ (\\lambda x. f(x)/g(x))\\ s$$$

Lean4

@[to_additive]
theorem div (hf : LipschitzWith Kf f) (hg : LipschitzWith Kg g) : LipschitzWith (Kf + Kg) fun x => f x / g x := by
  simpa only [div_eq_mul_inv] using hf.mul hg.inv

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