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Topic: Lipschitz continuity

Related Topics

Lipschitz maps

Continuous

Continuous function

Kirszbraun theorem

Tietze extension theorem

Short map

Ordinary differential equation

Metric space

Banach fixed point theorem

Discrete space

Rudolf Lipschitz

Rectifiable curve

Topological space

Compact space

Algebraic geometry

	Linear Approximations
		Thus there is a link between continuity and differentiability: If a function is differentiable at a point, it is also continuous there.
		Nevertheless, a function may be uniformly continuous without having a bounded derivative.
		is uniformly continuous on [0,1], but its derivative is not bounded on [0,1], since the function has a vertical tangent at 0.
	www.sosmath.com /calculus/diff/der10/der10.html (236 words)

	PlanetMath: Lipschitz condition and differentiability result
		About lipschitz continuity of differentiable functions the following holds.
		"Lipschitz condition and differentiability result" is owned by paolini.
		This is version 2 of Lipschitz condition and differentiability result, born on 2003-04-04, modified 2004-03-15.
	planetmath.org /encyclopedia/LipschitzConditionAndDifferentiabilityResult.html (78 words)

	Lipschitz continuity - ExampleProblems.com
		Lipschitz continuous maps with Lipschitz constant K = 1 are called short maps and with K < 1 are called contraction mappings if M=N; the latter are the subject of the Banach fixed point theorem.
		Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.
		A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0).
	www.exampleproblems.com /wiki/index.php/Lipschitz_condition (578 words)

	Kids.Net.Au - Encyclopedia > Lipschitz continuous (Site not responding. Last check: )
		Lipschitz continuous maps with Lipschitz contant K < 1 are called contractions; they are the subject of the Banach fixed point theorem.
		Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.
		A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure zero).
	www.kids.net.au /encyclopedia-wiki/li/Lipschitz_continuous (266 words)

	NationMaster - Encyclopedia: Lipschitz maps (Site not responding. Last check: )
		Lipschitz continuous maps with Lipschitz constant K = 1 are called short maps and with K < 1 are called contraction mappings when M=N also; the latter are the subject of the Banach fixed point theorem.
		If U is a subset of the metric space M and f : U → R is a Lipschitz continuous map, there always exist Lipschitz continuous maps M → R which extend f and have the same Lipschitz constant as f (see also Kirszbraun theorem).
		A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0).
	www.nationmaster.com /encyclopedia/Lipschitz-maps (365 words)

	NationMaster - Encyclopedia: Lipschitz (Site not responding. Last check: )
		Lipschitz gave his name to the Lipschitz continuity condition; in fact he worked in a broad range of areas.
		Lipschitz continuous maps with Lipschitz constant K = 1 are called short maps and with K
		This is version 2 of Lipschitz inverse mapping theorem, born on 2004-06-17, modified 2004-06-18.
	www.nationmaster.com /encyclopedia/Lipschitz (236 words)

	Lipschitz continuity
		In mathematics, a function f : M → N between metric spaces M and N is called Lipschitz continuous (or is said to satisfy a Lipschitz condition) if there exists a constant K > 0 such that d(f(x), f(y)) ≤ K d(x, y) for all x and y in M.
		Lipschitz continuity is an important condition in the existence and uniqueness theorem for ordinary differential equations.
		A Lipschitz continuous map f : I → R, where I is an interval in R, is almost everywhere differentiable (everywhere except on a set of Lebesgue measure 0).
	www.xasa.com /wiki/en/wikipedia/l/li/lipschitz_continuity.html (269 words)

	2. Decimal View of Limits
		Limits and continuity in calculus may be described geometrically, that is, intuitively and informally, or more precisely in terms of say epsilons and deltas.
		Here we may want to say for any k, there is an m such f(x) will agree with f(A) to k decimals if x agrees with A to m decimals.
		Now will say that f(x) is continuous at x = A if for each whole number k, there is a number m such the limit f(a) and the value of f(x) will agree to k-decimals whenever the number x agrees with the value of a to m decimal places.
	whyslopes.com /Calculus-Introduction/limits_technical_view.html (788 words)

	Content Guide
		For continuous functions, appendix F continues with statements and proofs of theorems
		Riemann Sums Convergence - a proof or two of the fundamental theorem of calculus in general and for base-2 Riemann Sums, based on the Equicontinuity Theorem for continuous integrands.
		Lipschitz Continuous Integrands and convergence of Riemann Sums.
	whyslopes.com /etc/Real-Analysis-Decimal-View (1769 words)

	H. Lipschitz Continuity II
		Most piecewise continuous functions met in practice, that is, in calculus courses, are Lipschitz continuous on each interval [a,b] which does not include a singularity - a point where division by zero occurs.
		In any event, Lipschitz continuous functions and criteria which identify them provide further examples of continuous functions and another link to error control or error bounding in computations.
		This author is undecided as to whether or not Lipschitz continuity should be emphasized in first courses on calculus.
	whyslopes.com /etc/Real-Analysis-Decimal-View/appH1_LIpshitz-Continuity-Integration.html (636 words)

	G. Lipschitz Continuity I
		Then f(x) is said to be Lipschitz Continuous on this interval [a,b] if and only if there is a constant K
		are both in the interval [a,b] then the number K is called a Lipschitz continuity constant for the function f(x) on the interval [a,b].
		Suppose K > 0 is a Lipschitz continuity constant for the function f(x) on the interval [a,b].
	whyslopes.com /etc/Real-Analysis-Decimal-View/appGb-Lipschitz-Continuity.html (517 words)

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