Editing Lipschitz continuity (section)

==Lipschitz manifolds==
A '''Lipschitz structure''' on a [[topological manifold]] is defined using an [[atlas (topology)|atlas of charts]] whose transition maps are bilipschitz; this is possible because bilipschitz maps form a [[pseudogroup]].  Such a structure allows one to define locally Lipschitz maps between such manifolds, similarly to how one defines smooth maps between [[smooth manifold]]s: if {{mvar|M}} and {{mvar|N}} are Lipschitz manifolds, then a function <math>f:M \to N</math> is '''locally Lipschitz''' if and only if for every pair of coordinate charts <math>\phi:U \to M</math> and <math>\psi:V \to N</math>, where {{mvar|U}} and {{mvar|V}} are open sets in the corresponding Euclidean spaces, the composition
<math display="block">\psi^{-1} \circ f \circ \phi:U \cap (f \circ \phi)^{-1}(\psi(V)) \to V</math>
is locally Lipschitz.  This definition does not rely on defining a metric on {{mvar|M}} or {{mvar|N}}.<ref name="Rosenberg">{{cite conference |first=Jonathan |last=Rosenberg |author-link=Jonathan Rosenberg (mathematician) |book-title=Miniconferences on harmonic analysis and operator algebras (Canberra, 1987) |title=Applications of analysis on Lipschitz manifolds |year=1988 |publisher=[[Australian National University]] |location=Canberra |pages=269–283 |url=https://projecteuclid.org/proceedings/proceedings-of-the-centre-for-mathematics-and-its-applications/Miniconference-on-Harmonic-Analysis-and-Operator-Algebras/Chapter/Applications-of-analysis-on-Lipschitz-manifolds/pcma/1416336222}} {{MathSciNet|id=954004}}</ref>

This structure is intermediate between that of a [[piecewise-linear manifold]] and a [[topological manifold]]: a PL structure gives rise to a unique Lipschitz structure.<ref>{{SpringerEOM|title=Topology of manifolds}}</ref>  While Lipschitz manifolds are closely related to topological manifolds, [[Rademacher's theorem]] allows one to do analysis, yielding various applications.<ref name="Rosenberg"/>