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...vative, one can generalize [[Rademacher's theorem]] to metric space-valued Lipschitz functions. ...ed by ''f''(''x'') = ''χ''_[0,''x''], this function is Lipschitz (and in fact, an [[isometry]]) since, if 0 ≤ ''x'' ≤ '' ...

...multiplicative constant. These maps are also related to [[quasiconformal]] maps, since in many circumstances they are in fact equivalent.<ref>{{cite book| ; Quasisymmetric maps preserve relative sizes of sets : If <math>A</math> and <math>B</math> are ...

== Types of maps == Just as there are various types of manifolds, there are various types of maps of manifolds. ...

In mathematics, a '''metric projection''' is a function that maps each element of a [[metric space]] to the set of points nearest to that ele ...losed and convex, then ''p_M'' is [[Lipschitz continuous]] with Lipschitz constant 1.{{citation needed|date= June 2024}} ...

The '''Piola transformation''' maps vectors [[Lagrangian and Eulerian specification of the flow field|between E ...et <math> K=F(\hat{K}) </math> with <math> \hat{K} </math> a domain with Lipschitz boundary. The mapping ...

...[[Brouwer fixed-point theorem]]: that is, <math>f</math> is continuous and maps the unit [[N-cube|''d''-cube]] to itself. The [[Brouwer fixed-point theorem ...Often, it is assumed that <math>f</math> is not only continuous but also [[Lipschitz continuous]], that is, for some constant <math>L</math>, <math>|f(x)-f(y)| ...

...> Gerald Jungck refined DeMarr's conditions, showing that they need not be Lipschitz continuous, but instead satisfy similar but less restrictive criteria.<ref> ...|date=1996 |title=Equivalent conditions involving common fixed points for maps on the unit interval |url=https://www.ams.org/journals/proc/1996-124-10/S00 ...

* The set of all [[Lipschitz continuity|Lipschitz curves]] (so that <math>\left\{ \dfrac{c(t) - c(s)}{t - s} : t \neq s {,} | * Any Lipschitz curve in <math>E</math> is locally Riemann integrable. ...

..., \mathrm{d} \mu(x) \right| f \colon X \to \mathbb{R} \text{ is bounded, 1-Lipschitz and has } \int_{X} f(x) \, \mathrm{d} \mu(x) = 0 \right\}.</math> The Laplace functional maps from the positive real line to the positive (extended) real line, or in mat ...

...ath>, we denote by <math>\mathrm{Lip}(f)</math> its [[Lipschitz continuity|Lipschitz seminorm]]: ...riples (and more generally, on seminorms which play a role of analogue for Lipschitz seminorms) for Connes' distance to indeed induce the weak* topology on the ...

...is to design an output feedback law <math>\pi:\, y \mapsto u</math> which maps the observed process <math>y</math> to the control input <math>u</math> in ===Lipschitz-continuous control laws=== ...

...ent the same transformation, so the '''versor group''' (also called the '''Lipschitz group''') is a [[double covering group|double cover]] of the conformal orth * The transformation maps concentric {{mvar|k}}-spheres to concentric {{mvar|k}}-spheres for every {{ ...

...S0002-9939-1995-1234626-0|title=The diameter conjecture for quasiconformal maps is true in space|year=1995|last1=Heinonen|first1=Juha|author-mask=2|journal * {{cite journal|doi=10.1007/BF02392747|title=Quasiconformal maps in metric spaces with controlled geometry|year=1998|last1=Heinonen|first1=J ...

{{defn|A set <math>S</math> of maps between fixed metric spaces is said to be [[equicontinuous]] if for each <m {{term|Lipschitz}} ...

...er the name "bracket polynomials". Since the theory is in the setting of [[Lipschitz function]]s, which are ''a fortiori'' continuous, the discontinuity of the The imaginary exponential function <math>e(x)</math> maps the real numbers to the circle group (see [[Euler's formula#Topological int ...

...s the image of an interval or the circle under a [[Lipschitz continuity|bi-Lipschitz map]] ''f'', i.e. satisfying ...hat the complement of the images of ''f'' and ''g'' is a Jordan curve. The maps ''f'' and ''g'' extend continuously to the circle |''z''| = 1 and the sewin ...

...en two vectors is the sum of their coordinate differences. If an embedding maps all pairs of vertices with distance <math>d</math> to pairs of vectors with | title = On Lipschitz embedding of finite metric spaces in Hilbert space ...

Note that the above defined Lyapunov dimension is invariant under Lipschitz [[diffeomorphisms]].<ref name=Kuznetsov-2016-PLA/><ref name=KuznetsovAL-201 | title=Sinai–Bowen–Ruelle measures for certain Henon maps ...

...e harmonic analysis of L.C. Young, the geometric algebra of K.T. Chen, the Lipschitz function theory of H. Whitney and core ideas of stochastic analysis. The co ...th>. Let <math>\mathbf{X}</math> and <math>\mathbf{Y}</math> be continuous maps <math>\triangle_{0,1}\to T^{(\lfloor p \rfloor)}(\mathbb{R}^{d})</math>. ...

* 1977 "Hausdorff content and rational approximation in fractional Lipschitz norms", Transactions AMS 228 (1977) 187-206. * 1986 "<math>C^\infty</math> maps may increase <math>C^\infty</math> dimension", Inventiones Math. 89 (1987) ...