• An everywhere differentiable function
g :
R →
R is Lipschitz continuous (with
K = sup |
g′(
x)|) if and only if it has a 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 m, where
U is an open set in
Rn, is
almost everywhere differentiable. Moreover, if
K is the best Lipschitz constant of
f, then \|Df(x)\|\le K whenever the
total derivative Df exists. • For a differentiable Lipschitz map f: U \to \R^m the inequality \|Df\|_{L^{\infty}(U)}\le K holds for the best Lipschitz constant K of f. If the domain U is convex then in fact \|Df\|_{L^{\infty}(U)}= K. • Suppose that {
fn} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all
fn have Lipschitz constant bounded by some
K. If
fn 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 {
fn} 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
fα with common constant, the function \sup_\alpha f_\alpha (and \inf_\alpha f_\alpha) 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 ::\tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\}, :where
k is a Lipschitz constant for
f on
U. ==Lipschitz manifolds==