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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 [a, b].
**Conversely, if f : I -> R is absolutely continuous and thus differentiable almost everywhere, and satisfies |f´(x)| <= K for almost all x in I, then f 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 f : U -> R^m, where U is an open set in Rⁿ, is almost everywhere differentiable. Moreover, if K is the best Lipschitz constant of f, then whenever the total derivative Df exists.
*For a differentiable Lipschitz map the inequality holds for the best Lipschitz constant of . If the domain is convex then in fact .
*Suppose that {f_n} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all f_n have Lipschitz constant bounded by some K. If f_n converges to a mapping f uniformly, then f 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 a subalgebra of the Banach space of continuous functions, and thus dense in it, an elementary consequence of the Stone–Weierstrass theorem (or as a consequence of Weierstrass approximation theorem, because every polynomial is locally Lipschitz continuous).
*Every Lipschitz continuous map is uniformly continuous, and hence continuous. More generally, a set of functions with bounded Lipschitz constant forms an equicontinuous set. The Arzelà-Ascoli theorem implies that if {f_n} 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).
*For a family of Lipschitz continuous functions f_? with common constant, the function (and ) is Lipschitz continuous as well, with the same Lipschitz constant, provided it assumes a finite value at least at a point.
*If U is a subset of the metric space M and f : U -> R is a Lipschitz continuous function, there always exist Lipschitz continuous maps M -> R that extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem). An extension is provided by
::
:where k is a Lipschitz constant for f on U.

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