Lipschitz Continuity - Properties

Properties

• An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has bounded first derivative; one direction follows from the mean value theorem. In particular, any continuously differentiable function is locally Lipschitz, as continuous functions are locally bounded so its gradient is locally bounded as well.

• A Lipschitz function g : R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero. Its derivative is essentially bounded in magnitude by the Lipschitz constant, and for a < b, the difference g(b) − g(a) is equal to the integral of the derivative g′ on the interval .
  • Conversely, if ƒ : I → R is absolutely continuous and thus differentiable almost everywhere, and satisfies |ƒ′(x)| ≤ K for almost all x in I, then ƒ is Lipschitz continuous with Lipschitz constant at most K.
  • More generally, Rademacher's theorem extends the differentiability result to Lipschitz mappings between Euclidean spaces: a Lipschitz map ƒ : U → Rm, where U is an open set in Rn, is almost everywhere differentiable. Moreover, if K is the best Lipschitz constant of ƒ, then whenever the total derivative Dƒ exists .
• For a differentiable Lipschitz map ƒ : U → Rm the inequality holds for the best Lipschitz constant of f, and it turns out to be an equality if the domain U is convex.
• Suppose that is a sequence of Lipschitz continuous mappings between two metric spaces, and that all have Lipschitz constant bounded by some K. If ƒ_n converges to a mapping ƒ uniformly, then ƒ is also Lipschitz, with Lipschitz constant bounded by the same K. In particular, this implies that the set of real-valued functions on a compact metric space with a particular bound for the Lipschitz constant is a closed and convex subset of the Banach space of continuous functions. This result does not hold for sequences in which the functions may have unbounded  Lipschitz constants, however. In fact, the space of all Lipschitz functions on a compact metric space is dense in the Banach space of continuous functions, an elementary consequence of the Stone–Weierstrass theorem.
• Every Lipschitz continuous map is uniformly continuous, and hence a fortiori continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà–Ascoli theorem implies that if is a uniformly bounded sequence of functions with bounded Lipschitz constant, then it has a convergent subsequence. By the result of the previous paragraph, the limit function is also Lipschitz, with the same bound for the Lipschitz constant. In particular the set of all real-valued Lipschitz functions on a compact metric space X having Lipschitz constant  ≤ K  is a locally compact convex subset of the Banach space C(X).
• If U is a subset of the metric space M and ƒ : U → R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M → R which extend ƒ and have the same Lipschitz constant as ƒ (see also Kirszbraun theorem). An extension is provided by where k is a Lipschitz constant for ƒ on U.