Editing Lipschitz continuity (section)

==Properties==
*An everywhere differentiable function ''g''&nbsp;:&nbsp;'''R'''&nbsp;→&nbsp;'''R''' is Lipschitz continuous (with ''K''&nbsp;=&nbsp;sup&nbsp;|''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''&nbsp;:&nbsp;'''R'''&nbsp;→&nbsp;'''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''&nbsp;< ''b'', the difference ''g''(''b'')&nbsp;−&nbsp;''g''(''a'') is equal to the integral of the derivative ''g''′ on the interval [''a'',&nbsp;''b''].
**Conversely, if ''f''&nbsp;: ''I''&nbsp;→ '''R''' is absolutely continuous and thus differentiable almost everywhere, and satisfies |''f′''(''x'')|&nbsp;≤ ''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''&nbsp;:&nbsp;''U''&nbsp;→&nbsp;'''R'''<sup>''m''</sup>, where ''U'' is an open set in '''R'''<sup>''n''</sup>, is [[almost everywhere]] [[derivative|differentiable]]. Moreover, if ''K'' is the best Lipschitz constant of ''f'', then <math>\|Df(x)\|\le K</math> whenever the [[total derivative]] ''Df'' exists.{{citation needed|date=March 2023}}
*For a differentiable Lipschitz map <math>f: U \to \R^m</math> the inequality <math>\|Df\|_{W^{1,\infty}(U)}\le K</math> holds for the best Lipschitz constant <math>K</math> of <math>f</math>. If the domain <math>U</math> is convex then in fact <math>\|Df\|_{W^{1,\infty}(U)}= K</math>.{{Explain|date=November 2019}}
*Suppose that {''f<sub>n</sub>''} is a sequence of Lipschitz continuous mappings between two metric spaces, and that all ''f<sub>n</sub>'' have Lipschitz constant bounded by some ''K''.  If ''f<sub>n</sub>'' converges to a mapping ''f'' [[uniform convergence|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&ndash;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 function|continuous]]. More generally, a set of functions with bounded Lipschitz constant forms an  [[equicontinuous]] set.  The [[Arzelà–Ascoli theorem]] implies that if {''f<sub>n</sub>''} 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 ≤&nbsp;''K''&thinsp; is a [[Locally compact space|locally compact]] convex subset of the Banach space ''C''(''X'').
*For a family of Lipschitz continuous functions ''f''<sub>α</sub> with common constant, the function <math>\sup_\alpha f_\alpha</math> (and <math>\inf_\alpha f_\alpha</math>) 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''&nbsp;: ''U''&nbsp;→ '''R''' is a Lipschitz continuous function, there always exist Lipschitz continuous maps ''M''&nbsp;→ '''R''' that extend ''f'' and have the same Lipschitz constant as ''f'' (see also [[Kirszbraun theorem]]). An extension is provided by 
::<math>\tilde f(x):=\inf_{u\in U}\{ f(u)+k\, d(x,u)\},</math> 
:where ''k'' is  a Lipschitz constant for ''f'' on ''U''.