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Topic: Uniform continuity

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	PlanetMath: uniform continuity of Lipschitz functions
		Proposition 1 A Lipschitz mapping is uniformly continuous.
		"uniform continuity of Lipschitz functions" is owned by paolini.
		This is version 4 of uniform continuity of Lipschitz functions, born on 2005-02-28, modified 2006-07-18.
	www.planetmath.org /encyclopedia/UniformContinuityOfLipschitzFunctions.html (59 words)

	Uniform continuity - Wikipedia, the free encyclopedia
		In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) ("continuity"), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself ("uniformity").
		Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and when we speak of a function being continuous on an interval, we mean only that it is continuous at each point of the interval.
		A function either is uniformly continuous on an entire interval or is not; it may be continuous at each point of an interval without being uniformly continuous on the entire interval.
	en.wikipedia.org /wiki/Uniform_continuity (465 words)

	Uniform continuity - Definition, explanation
		In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x effect small changes in the output f(x) ("continuity"), and furthermore the size of the changes in f(x) depends only on the size of the changes in x but not on x itself ("uniformity").
		Continuity itself is a local property of a function—that is, a function f is continuous, or not, at a particular point, and when we speak of a function being continuous on an interval, we mean only that it is continuous at each point of the interval.
		A function is uniformly continuous, or not, on an entire interval, and may be continuous at each point of an interval without being uniformly continuous on the entire interval.
	www.calsky.com /lexikon/en/txt/u/un/uniform_continuity.php (485 words)

	PlanetMath: uniformly continuous
		A more general definition of uniform continuity applies to functions between metric spaces (there are even more general environments for uniformly continuous functions, i.e.
		Uniformly continuous functions have the property that they map Cauchy sequences to Cauchy sequences and that they preserve uniform convergence of sequences of functions.
		This is version 11 of uniformly continuous, born on 2002-06-07, modified 2006-09-21.
	planetmath.org /encyclopedia/UniformlyContinuousFunction.html (217 words)

	The Heine-Borel Theorem
		The Heine-Borel theorem is used in the theory of uniform continuity and uniform convergence.
		This is called uniform continuity, and we have shown that a function continuous in a closed interval is uniformly continuous in that interval.
		Uniform continuity does not necessarily follow from continuity over an open interval, however, as we have seen.
	www.du.edu /~etuttle/math/heinebo.htm (1294 words)

	6.2. Continuous Functions
		Continuous functions can be added, multiplied, divided, and composed with one another and yield again continuous functions.
		Continuity is defined at a single point, and the epsilon and delta appearing in the definition may be different from one point of continuity to another one.
		The difference is that the delta in the definition of uniform continuity depends only on epsilon, whereas in the definition of simply continuity delta depends on epsilon as well as on the particular point c in question.
	www.shu.edu /projects/reals/cont/contin.html (734 words)

	Springer Online Reference Works
		The properties of proximity spaces are a generalization of the uniform properties of a metric space —; in analogy to the generalization of the latter's continuity properties to a topological space.
		-continuity in the sense of the latter is equivalent to uniform continuity [2]; the proximity spaces for which such a metric is possible are said to be metrizable.
		The set of uniform coverings of a proximity space is identical with the union of all uniform structures compatible with this space [4].
	eom.springer.de /p/p075560.htm (1719 words)

	[No title]
		Uniform convergence of martingales in the one-dimensional branching random walk.
		It is therefore plain that uniform convergence implies pointwise convergence.
		The examples of continuous nowhere differentiable functions given in most analysis texts involve the uniform convergence of a series of functions.
	www.lycos.com /info/uniform-convergence.html (337 words)

	Talk - Crumpled Paper
		While all of the continuity arguments are supplemented with large doses of intuition, a listener who has not at least seen continuity of real, one-variable functions before will struggle with this talk.
		Knowledge of the epsilon-delta definition of continuity is useful but not necessary.
		There is also some uniform continuity hand-waving towards the end of the talk, my assumption being that much of the audience will not understand uniform continuity.
	www.amherst.edu /~sgoldstine/exposition/bro.html (326 words)

	Uniform Calculus
		Uniform continuity of a one variable function f is a condition on its variation, f(y) - f(x).
		A composition of uniformly continuous functions is uniformly continuous.
		A uniformly continuous function f on a finite interval I is bounded.
	mystic.math.neu.edu /bridger/lbc1a/node2.html (220 words)

	AFSO21 Leans Out Uniform Development
		"The uniform process is definitely an enterprise system, and by that I mean there are multiple players who participate in the uniform process," he said.
		Another discovery was the lack of continuity between uniform boards and the data they create.
		Even with the completion of those final events, the uniform development process will not be completely "optimized." Part of AFSO21 is "continual process improvement." That means there always will be a need to look deeper into a process to eliminate more waste and inefficiency.
	www.military.com /features/0,15240,110399,00.html?ESRC=airforcenews.RSS (915 words)

	ROI calculations a rarity in business continuity planning
		The results of Continuity Central’s survey into whether a return on investment calculation is made as a part of the BC process.
		One of the most common questions asked through Continuity Central's FAQ service is how to calculate the return on investment made in business continuity activities and processes.
		Business continuity is at its most successful when it spends a small amount to demonstrably prevent the loss of a much larger amount.
	www.continuitycentral.com /feature034.htm (1438 words)

	Sunshine School Uniforms
		If a student needs a garment that is outside the wide range of sizes available, we can make a “special made” garment (there is a small, extra charge per garment as this is cut and sewn to your specific measurements).
		Continuity = being able to obtain the same style, color, cut on an ongoing basis.
		Many Sunshine garments have special soil and stain release treatments and growth features that extend the life of a garment by enabling you to let down a generous hem or let out a specially-designed waistband.
	www.sunshineuniforms.com /faq.htm (960 words)

	Karl's Calculus Tutor - The Nitty Gritty of the Fundamental Theorem
		being continuous is enough to guarantee what we need for any rectangle in particular, but not enough for the infinite crowd of them that we end up with when we take the limit.
		There is a stronger form of continuity called uniform continuity that is required for the Riemann sum to be guaranteed a limit.
		There is a theorem (whose proof is well beyond first year students) that when a domain interval is closed and bounded (that is it includes the endpoints and has endpoints at both ends), then any function that is continuous on that entire interval is also uniformly continuous on that entire interval.
	www.karlscalculus.org /l10_2a.html (780 words)

	Analysis WebNotes: Chapter 06, Class 33
		We'll start by re-examining the definition of continuity for a function between two metric spaces.
		From this point of view, it is clear that one thing which could cause a continuous function to fail to be uniformly continuous would be is the slope of the line becomes too large.
		The following result shows why uniform continuity is in fact a very common property, despite being, apparently, much stronger than simple continuity.
	www.math.unl.edu /~webnotes/classes/class33/class33.htm (364 words)

	MATH 251 FINAL EXAM INFORMATION (Site not responding. Last check: 2007-10-10)
		Theorem that the uniform limit of a sequence of continuous functions is continuous.
		Theorem that a uniformly Cauchy sequence of functions is guaranteed to converge uniformly to a function.
		Theorem about the existence of a max and min for a real valued function which is continuous on a closed and bounded set (interval).
	www.math.wvu.edu /~sherm/m251/final.remarks.html (421 words)

	Mathematics Other Homework Help
		Prove (or disprove) the following statement: A function f exists that is uniformly continuous on (a,oo) and for which lim as x-> oo of f(x) = oo (infinity symbol).
		I know that f(x) = x ^ (1/3) (cube root of x) is uniformly continuous on R, and that it's limit as x approaches infinity is infinite.
		Proof of Uniform Continuity - (See attached file for full problem description and equations) --- Show that the function is uniformly continuous on.
	www.brainmass.com /homeworkhelp/math/other/56352 (291 words)

	Analysis WebNotes: Contents Page
		Continuous functions are bounded on sequentially compact sets
		A continuous bijection from a sequentially compact set has a continuous inverse
		Continuous functions attain their maximum on sequentially compact sets
	www.math.unl.edu /~webnotes/contents/chapters.htm (190 words)

	Dempsey - Uniform & Linen Supply
		The uniform and linen supply business is about more than delivering products.
		It was a logical outgrowth of their father's towel and coverall supply business which was established thirty-six years earlier.
		On a per uniform basis (1 shirt and 1 pant), commercial laundering uses 64% less water and 73% less energy than home washing.
	www.dempseyuniform.com /about.php (415 words)

	FFIEC Home Page
		The online version of the UBPR includes 5 years of continuous financial history that is restated quarterly to reflect call report amendments and new analytical tools.
		The FFIEC announced that Uniform Bank Performance Reports (UBPR) are now available in electronic format through the FFIEC website at www.ffiec.gov in the Information Services section.
		The Uniform Retail Credit Classification and Account Management Policy published in today's Federal Register updates and expands the classification policy for retail credit loans that was issued in 1980.
	www.ffiec.gov /press.htm (5917 words)

	Uniform Continuity 2
		The difference is that, for uniform continuity, the same
		That's essentially what you are saying since you seem to be using a slightly different (equivalent) definition of uniform continuity.
		Assuming f(a) is not a and f(b) is not b, I've basically proved that a continuous function defined on [a,b] must intersect the identity function g(x)=x at least once.
	www.physicsforums.com /showthread.php?t=97677 (416 words)

	Sample questions
		Give an example of a function which is continuous, but not uniformly continuous.
		Prove that your function does not satisfy the definition of uniform continuity.
		Prove that f is continuous at x=1/2 and discontinuous everywhere else.
	www.math.mtu.edu /graduate/prof/node9.html (290 words)

	[No title]
		and constant functions are continuous (pf.); - Operations preserving continuity: restriction, +,-,x, and composition of fn.
		(pf.) - The extension problem: cluster points, limit of a function at a cluster point; - Continuity rephrased using “the limit of a function” concept; - Sequential continuity of a function: definition & equivalence to continuity of a function; - Projections are continuous fns.
		integration & limit commute; - Differentiation & limit: the problem, poitwise convergence is not enough (example — see continuity!) - Uniform convergence of derivatives + convergence at a point => differentiation & limit commute; - Infinite series: def.; properties: translating properties of convergence of sequences; - Operations with series: translating limit laws for sequences (pf.
	www.ilstu.edu /~lmiones/347rvs04.doc (1596 words)

	Mathematics Other Homework Help
		Also please use the definition of uniform continuity in Real Analysis.
		Uniform Continuity - Prove (or disprove) the following statement: A function f exists that is uniformly continuous on (a,oo) and for which lim as x-> oo of f(x) = oo (infinity symbol).
		Real Analysis--Uniformly Continuous Functions - Determine whether or not each of the following functions is uniformly continuous on the given set D. Give reasons to your answers.
	www.brainmass.com /homeworkhelp/math/other/35500 (301 words)

	Uniform Continuity (Site not responding. Last check: 2007-10-10)
		A function f is uniformly continuous, or uniform, throughout a region R if one δ fits all.
		Every uniform function is continuous, but the converse is not true.
		This function, and any other function that approaches infinity, is not uniform.
	www.mathreference.com /lc,uni.html (108 words)

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