For a Lipschitz continuous function, there exists a double cone (white) whose origin can be moved along the graph so that the whole graph always stays outside the double cone
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number; the smallest such bound is called the Lipschitz constant of the function (and is related to the modulus of uniform continuity). For instance, every function that is defined on an interval and has a bounded first derivative is Lipschitz continuous.[1]
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem.[2]
We have the following chain of strict inclusions for functions over a closed and bounded non-trivial interval of the real line:
mathematical analysis, Lipschitzcontinuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively...
= 1 {\displaystyle \alpha =1} is referred to as Lipschitzcontinuity. That is, a function is Lipschitz continuous if there is a constant K such that the...
the same modulus of continuity are exactly equicontinuous families. For instance, the modulus ω(t) := kt describes the k-Lipschitz functions, the moduli...
: R n → R m {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} ^{m}} is Lipschitz continuous (with the appropriate normed spaces as needed) in the neighbourhood...
absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows...
describe a function that satisfies the Lipschitz condition, a strong form of continuity, named after Rudolf Lipschitz. The surname may refer to: Daniel Lipšic...
{\displaystyle L} is uniformly continuous, and even Lipschitz continuous. Conversely, it follows from the continuity at the zero vector that there exists a ε >...
However, any Lipschitz map between metric spaces is uniformly continuous, in particular any isometry (distance-preserving map). Although continuity can be defined...
the endpoints of the curves and both of which obey a Lipschitzcontinuity condition with Lipschitz constant less than one. Tao also formulated several...
Rudolf Lipschitz. Let f be a function on [0,2π], let t be some point and let δ be a positive number. We define the local modulus of continuity at the...
local Lipschitz constant for the gradient ∇ f {\displaystyle \nabla f\,} near the point x {\displaystyle \mathbf {x} } (see Lipschitzcontinuity). If the...
of a function Uniform continuity Modulus of continuityLipschitzcontinuity Semi-continuity Equicontinuous Absolute continuity Hölder condition – condition...
For linear operators between normed vector spaces, Lipschitzcontinuity is equivalent to continuity—an operator satisfying either of these conditions is...
h} is an example of a function that is differentiable but not locally Lipschitz continuous. The exponential function e x {\displaystyle e^{x}} is analytic...
well as uniform, Lipschitz, and Hölder continuity, can be defined in the setting of metric spaces. Other notions, such as continuity, compactness, and...
assumes more and concludes more. It requires Lipschitzcontinuity, while the Peano theorem requires only continuity; but it proves both existence and uniqueness...
topology and algebraic geometry. The Hirzebruch–Riemann–Roch theorem, Lipschitzcontinuity, the Petri net, the Schönhage–Strassen algorithm, Faltings's theorem...
mathematical analysis, Dini continuity is a refinement of continuity. Every Dini continuous function is continuous. Every Lipschitz continuous function is...
unconventional value for sgn(0).) The signum function does not satisfy the Lipschitzcontinuity condition required for the usual theorems guaranteeing existence...
study was mainly focused on the study of differential equations and Lipschitzcontinuity. some of his notable contributions include: On the Composition Operator...
are special cases of metric spaces, and thus one has a notion of Lipschitzcontinuity, Hölder condition, together with a coarse structure, which leads...
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