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Topic: Uniformly continuous

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	PlanetMath: uniformly continuous
		Every uniformly continuous function is also continuous, while the converse does not always hold.
		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)

	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/Uniformly_continuous (465 words)

	The Heine-Borel Theorem
		This is called uniform continuity, and we have shown that a function continuous in a closed interval is uniformly continuous in that interval.
		It is not uniformly continuous in this interval, because δ → 0 as we approach the origin.
		Indeed, 1/x is uniformly continuous in a < x < b, with b > a > 0.
	www.du.edu /~etuttle/math/heinebo.htm (1294 words)

	PlanetMath: contractive maps are uniformly continuous
		"contractive maps are uniformly continuous" is owned by mathcam.
		Cross-references: metric, metric space, uniformly continuous, contraction mapping
		This is version 3 of contractive maps are uniformly continuous, born on 2003-07-19, modified 2004-04-02.
	planetmath.org /encyclopedia/ContractiveMapsAreUniformlyContinuous.html (83 words)

	Continuity
		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.
	pirate.shu.edu /projects/reals/cont/contin.html (733 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)

	Piecewise Continuous (Site not responding. Last check: 2007-10-20)
		Thus it is sufficient to consider g, a function that is continuous over part of the domain from 0 to 2π, and zero elsewhere.
		continuous on a closed interval implies uniformly continuous.
		Continuous functions can still be very wiggly at the microscopic level, like the function that is everywhere continuous and nowhere differentiable.
	www.mathreference.com /la-xf-four,pcont.html (337 words)

	Limit and Continuity
		Continuity for f (x) at x = c (both side continuity).
		otherwise f (x) is not continuous at x = c.
		is uniformly continuos over [ 0, 4 ], since [0, 4] is a compact set.
	www.usd.edu /~ylio/Calcul/Limit_and_Continuity.html (736 words)

	Information about the final exam in mathematics 328 (Site not responding. Last check: 2007-10-20)
		Prove that a continuous function defined on a compact set is necessarily uniformly continuous.
		Prove that the limit of a uniformly converging sequence of continuous funtions is continuous.
		(x) converge pointwise and are differentiable on a certain interval, and if their derivatives converge uniformly on that interval, then the limit function f(x) is differentiable on the interval, and the derivative of f is the limit of the derivatives of the functions in the sequence.
	www.mast.queensu.ca /~leo/essays.html (644 words)

	Two Proofs of the Cauchy-Peano Theorem
		is a continuous function on a compact set).
		(continuous and has, perhaps, a jump discontinuity in the derivative at the partition points).
		is a consequence of the (uniform) continuity of
	www.math.unl.edu /~s-bbockel1/933-notes/Two_Proofs.html (199 words)

	Linear Approximations
		This relationship between differentiability and continuity is local.
		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)

	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)

	Topological vector space (Site not responding. Last check: 2007-10-20)
		A vector space is an abelian group with respect to the operation of addition, and in a topological vector space the inverse operation is always continuous (since it is the same as multiplication by -1).
		Vector addition and scalar multiplication are not only continuous but even homeomorphic which means we can construct a base for the topology and thus reconstruct the whole topology of the space from any local base around 0.
		Bornological space: a locally convex space where the continuous linear operators to any locally convex space are exactly the bounded linear operators.
	topological-vector-space.iqnaut.net (957 words)

	Continuity
		First problem is peculiar in that proving "bounded" is far simpler than proving "uniformly continuous".
		If you are working with "uniformly continuous" you should long ago have seen the proof that any continuous function (you don't need "periodic") is bounded.
		In order to prove uniformly continuous you will need to use the fact that a function continous on a compact set (here, closed and bounded) is uniformly continuous.
	www.physicsforums.com /showthread.php?t=72168 (475 words)

	Uniform Convergence
		This is similar to uniform continuity, where one δ fits all.
		If the range space is complex, or a real vector space, the sequence of functions is uniformly convergent iff all the component functions are uniformly convergent.
		Each function in the sequence is uniformly continuous, yet the limit function isn't even continuous.
	www.mathreference.com /lc-ser,unic.html (482 words)

	Sample questions
		Give an example of a function which is continuous, but not uniformly continuous.
		Prove that f is continuous at x=1/2 and discontinuous everywhere else.
		Prove that any continuous function f is Riemann integrable on [a,b].
	www.math.mtu.edu /graduate/prof/node9.html (290 words)

	Mathematics Other Homework Help
		Determine whether the following functions are uniformly continuous on the indicated intervals.
		Proving that f is not uniformly continuous - I need help with writing proofs for the two functions below.
		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/31010 (212 words)

	: Problem 4.52
		This function is uniformly continuous, even though it is unbounded.
		But, x and p also satisfy (), since f is uniformly continuous** and S is bounded.
		Note that f is continuous everywhere that it is defined, and it is defined on (0,1).
	home.cc.umanitoba.ca /cgi-bin/discus/board-auth.cgi?file=/10/50.html (558 words)

	Cauchy.html
		It is also the case that Cauchy sequences are not preserved under mapping by continuous functions.
		There is an important sub-class of continuous functions which do preserve Cauchy sequences and, in fact, are the continous functions on an important sub-Category of Complete Metric Spaces.
		To verify that complete metric spaces and uniformly continuous maps form a category we need to check the the composition of uniformly continuous maps is uniformly continuous.
	www.umsl.edu /~siegel/SetTheoryandTopology/Cauchy.html (236 words)

	No Title
		Show that any continuous map on the closed interval [0,1] is uniformly continuous.
		What is the difference between being uniformly continuous and continuous.
		Inverse image of a compact set under a continuous map is always compact.
	www.math.sunysb.edu /~myonghi/OW/Y2000/m5320/m5320.html (236 words)

	: Problem 4.51
		continuous on R. Have to prove that there is an e >0 such that for every d >0,
		Then f is uniformly continuous because for any e>0, take d=1/2.
		To summarize, (a) If we take S to be N, then f is uniformly on S. (b) f is uniformly continuous on (0,infinity) because it is clearly uniformly continuous on [0,2),
	home.cc.umanitoba.ca /cgi-bin/discus/board-auth.cgi?file=/10/47.html (281 words)

	Continuous Function Becomes Uniform (Site not responding. Last check: 2007-10-20)
		Let f be continuous on a closed bounded region r in n dimensions and suppose it is not uniform.
		Since f is continuous at p, let δ define a neighborhood about p such that f(x) is within ε/2 of f(p).
		There is a small region containing p, entirely inside this δ neighborhood, that is continuous, but not uniform.
	www.mathreference.com /lc,cuc.html (178 words)

	Uniformly connected space - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics a uniformly connected space or Cantor connected space is a uniform space U so that every uniformly continuous functions from U to a discrete uniform space is constant.
		A uniform space U is called uniformly disconnected if every uniformly continuous functions from a discrete uniform space to U constant.
		A compact topological space is uniformly connected if and only if it is connected
	en.wikipedia.org /wiki/Uniformly_connected_space (130 words)

	Brouwer's Cambridge Lectures on Intuitionism
		Consequently the science of classical (Euclidean, three-dimensional) space had to continue its existence as a chapter without priority, on the one hand of the aforesaid (exact) science of numbers, on the other hand (as applied mathematics) of (naturally approximative) descriptive natural science.
		Striking examples are the modern theorems that the continuum does not split, and that a full function of the unit continuum is necessarily uniformly continuous.
		There continued to reign some conviction that a mathematical assertion is either false or true, whether we know it or not, and that after the extinction of humanity mathematical truths, just as laws of nature, will survive.
	www.marxists.org /reference/subject/philosophy/works/ne/brouwer.htm (2524 words)

	Functions with distant fibers and uniform continuity by Alessandro Berarducci, Dikran Dikranjan and Jan Pelant (Site not responding. Last check: 2007-10-20)
		The uniformly approachable functions introduced in [DP] are defined by a property stronger than continuity and weaker than uniform continuity, which is preserved under composition.
		To prove this we show that every real valued continuous functions with distant fibers on a uniformly locally connected metric space is uniformly approachable, and any (weakly) uniformly approachable function on R
		the functions with many uniformly continuous truncations coincide with the functions with distant connected components of fibers.
	at.yorku.ca /i/d/e/b/34.htm (339 words)

	Real Analysis: Uniformly continuous
		Suppose f is a continuous real valued function on R - real #s and that 0
		b) Suppose the continuity assumption is left out, but the function f is still assumed to have 0
		By teh (second?) Fundamental Theorem of Calculus, F, in part a), is a differentiable function with (uniformly) bounded derivative.
	www.physicsforums.com /showthread.php?t=113869 (221 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-20)
		The composite of uniformly-continuous mappings is uniformly continuous.
		Uniform continuity of mappings occurs also in the theory of topological groups.
		The notion of uniform continuity has been generalized to mappings of uniform spaces (cf.
	eom.springer.de /U/u095220.htm (168 words)

	Mathematics Other Homework Help
		Let X be a continuous random variable having cumulative distribution function F. Define the random variable Y by Y = F(X).
		Show that Y is uniformly distributed over (0,1).
		Jones figures that the total number of thousands of miles that an auto can be driven before it would need to be junked as an exponential random variable with parameter 1/20.
	www.brainmass.com /homeworkhelp/math/other/34870 (225 words)

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