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Topic: Continuous function

	Continuous Function
		In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output.
		A continuous function with a continuous inverse function is called bicontinuous.
		Theorem 2.1 Assume g is a continuous function and pn from n=0 to n=infinity is a sequence determined by a fixed point iteration.
	www.lycos.com /info/continuous-function.html (384 words)

	Site Map
		Continuous Distributions - Beta Distribution - Moments Uncent.
		Continuous Distributions - Cauchy 1 (Parameter) Distribution - Distribution Function
		Continuous Distributions - Chi Square 1 Distribution - Related Distributions 13 - Chi Square 1-Parameter Distribution versus Double Exponential or Laplace Distribution
	www.xycoon.com /toc.htm (3938 words)

	Will the real continuous function please stand up?
		In everyday speech, a 'continuous' process is one that proceeds without gaps of interruptions or sudden changes.
		With limits defined in this way, the resulting definition of a continuous function is known as the Cauchy-Weierstrass definition, after the two nineteenth century mathematicians who developed it.
		The continuous function is formed by motion, which takes place over time.
	www.maa.org /devlin/devlin_11_06.html (1099 words)

	Mathematics - Apronus.com
		We intend our proof to be understandable for everyone who has basic familiarity with integer numbers and who is capable of concentrating his attention.
		Heine continuity implies Cauchy continuity without the Axiom of Choice - On this page we state and prove that every Heine continuous real function is also Cauchy continuous.
		Topological Methods In Real Analysis - proofs of the existence of a continuous nowhere differentiable function and a continuous nowhere monotonic function.
	www.apronus.com /math/math.htm (406 words)

	S.O.S. Mathematics CyberBoard :: View topic - Continuous Function
		Well, the range can be continuous if you take the square root to be plus or minus.
		Well, the [function] can be continuous if you take the square root to be plus or minus.
		well, the way you defined your function f (the domain being R) then f does not have an inverse because it is not one to one....
	www.sosmath.com /CBB/viewtopic.php?t=18713&highlight= (721 words)

	A function continuous at all irrationals, discontinuous at all rationals. (Site not responding. Last check: )
		A function continuous at all irrationals, discontinuous at all rationals.
		(I've sometimes heard this called the ``ruler'' function, since its graph vaguely resembles the markings on a ruler.) Then f has the surprising property that it is continuous at all irrationals and discontinuous at all rationals.
		It's a bit harder to see that f is continuous at any irrational x.
	www.math.tamu.edu /~tom.vogel/gallery/node6.html (253 words)

	Continuous function - Gurupedia
		In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output.
		Cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper.
		For example, if a child undergoes continuous growth from 1m to 1.5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have equalled 1.25m.
	www.gurupedia.com /c/co/continuous.htm (1000 words)

	NationMaster - Encyclopedia: Continuous function (topology) (Site not responding. Last check: )
		Dually, for a function f from a set S to a topological space, one defines the initial topology on S by letting the open sets of S be those subsets A of S for which f(A) is open in X.
		In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output.
		is a derivative of a continuous function in
	www.nationmaster.com /encyclopedia/Continuous-function-%28topology%29 (2823 words)

	Kids.Net.Au - Encyclopedia > Continuous function
		In mathematics, a continuous function is one in which "small" changes in the input produce "small" changes in the output.
		Such a function can be represented by a graph in the cartesian plane; the function is continuous if, roughly speaking, the graph is a single unbroken curve with no "holes" or "jumps": if it can be drawn by hand without lifting the pencil from the paper.
		A continuous map that is bijective such that its inverse map[?] is also continuous is called a homeomorphism.
	www.kids.net.au /encyclopedia-wiki/co/Continuous_function (1119 words)

	NationMaster - Encyclopedia: Continuity correction (Site not responding. Last check: )
		continuous function (topology) In topology and related areas of mathematics a continuous function is a morphism between topological spaces; that is, a mapping which preserves the topological structure.
		Continuity equations are the (stronger) local form of conservation laws In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves.
		In electromagnetic theory, the continuity equation is derived from two of Maxwell's equations Maxwell's equations are the set of four equations, attributed to James Clerk Maxwell (written by Oliver Heaviside), that describe the behavior of both the electric and magnetic fields, as well as their interactions with matter.
	www.nationmaster.com /encyclopedia/Continuity-correction (595 words)

	Science Fair Projects - Continuous function
		Continuity is particularly a concern in the production of film and television due to the difficulty of rectifying an error in continuity after shooting has completed, although it also applies to other art forms, including novels, comics and animation, though usually on a much broader scale.
		Many continuity errors are subtle, such as changes in the level of drink in a character's glass or the length of a cigarette, others can be more noticeable, such as changes in the clothing of a character.
		Care towards continuity must be taken because films are rarely filmed in the order they are presented in: that is, a crew may film a scene from the end of a movie first, followed by one from the middle, and so on.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Continuous (1214 words)

	PlanetMath: uniformly continuous (Site not responding. Last check: )
		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.
	www.planetmath.org /encyclopedia/UniformlyContinuous.html (218 words)

	SparkNotes: Functions, Limits, and Continuity: Continuity
		Formally, a function is continuous at a point x = c if the (standard two-sided) limit exists there and is equal to the value of the function at c.
		One problem that you might have to deal with is using the formal definition of continuity to determine whether a piecewise-defined function is continuous.
		The intermediate value theorem says that if f is continuous on the closed interval [a, b], then f attains each of the values between f (a) and f (b) at least once on the open interval (a, b).
	www.sparknotes.com /math/calcab/functionslimitscontinuity/section3.rhtml (576 words)

	Karl's Calculus Tutor: 3.1 More Continuity: Track to the Future
		Since the new function is defined simply as the sum of two continuous functions, and since continuity is defined in terms of their limits, it follows immediately that the sum of two continuous functions is continuous.
		If the two functions are continuous in some places but not in others, then their sum is continuous everywhere where BOTH functions are simultaneously continuous (note that the sum might be continuous at a point where both summands are not, but it doesn't have to be.
		Likewise, if the two functions are continuous in some places and not in others, then their quotient is continuous everywhere that they BOTH simultaneously continuous AND the denominator function is not zero (the analogous rule concerning sufficiency and necessity applies here as well, however, the quotient will never be continuous wherever the denominator is zero).
	www.karlscalculus.org /calc3_1.html (2529 words)

	SparkNotes: Functions, Limits, and Continuity: Terms
		A continuous function is one that is continuous for all points in its domain.
		Intermediate Value Theorem - If f is a continuous function on a closed interval [a, b], then for every value r that lies between f (a) and f (b), there exists a constant c on (a, b) such that f (c) = r.
		Odd Function - This is a function f for which f (- x) = - f (x) for all x in the domain.
	www.sparknotes.com /math/calcab/functionslimitscontinuity/terms.html (979 words)

	Continuous function (topology) - Definition, explanation
		In topology and related areas of mathematics a continuous function is a morphism between topological spaces, that is a mapping which preserves the topological structure.
		Surjective continuous functions between topological spaces are only possible if the topology of the codomain space is weaker than the topology of the domain space.
		In real analysis continuity of functions is commonly defined using the ε-δ definition; which builds on the property of the real line being a metric space.
	www.calsky.com /lexikon/en/txt/c/co/continuous_function__topology_.php (795 words)

	Continuous function - Definition, explanation
		More generally, we say that a function is continuous on some subset of its domain if it is continuous at every point of that subset.
		For example, if a child undergoes continuous growth from 1m to 1.5m between the ages of 2 years and 6 years, then, at some time between 2 years and 6 years of age, the child's height must have been 1.25m.
		As a consequence, if f is continuous on [a, b] and f(a) and f(b) differ in sign, then, at some point c, f(c) must equal zero.
	www.calsky.com /lexikon/en/txt/c/co/continuous_function.php (1180 words)

	6.2. Continuous Functions
		Continuous functions are precisely those groups of functions that preserve limits, as the next proposition indicates:
		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.
	web01.shu.edu /projects/reals/cont/contin.html (734 words)

	1.3.6.2. Related Distributions
		For a continuous function, the probability density function (pdf) is the probability that the variate has the value x.
		The hazard function is the ratio of the probability density function to the survival function, S(x).
		The cumulative hazard function is the integral of the hazard function.
	www.itl.nist.gov /div898/handbook/eda/section3/eda362.htm (597 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: )
		Well there are really only two kinds of functions that you will have to analyze for continuity, rational functions in which there is a fraction and the variable is in the denominator, and piecewise functions.
		Consider the function f(x) = sqrt(x-5) (Sqrt stands for square root - I can't type the radical symbol into this e-mail.) In this case, f is defined for all real numbers greater than or equal to 5 since you can't take the square root of a negative number.
		To summarize, continuous functions are functions in which the graphs are smooth curves or lines without breaks in them.
	mathforum.org /library/drmath/view/53745.html (797 words)

	continuous function \| English \| Dictionary & Translation by Babylon
		In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output.
		An intuitive though imprecise (and inexact) idea of continuity is given by the common statement that a continuous function is a function whose graph can be drawn without lifting the chalk from the flboard.
		All additive functions (functions which preserve all lubs) are continuous.
	www.babylon.com /definition/continuous_function (169 words)

	PlanetMath: continuous (Site not responding. Last check: )
		A related notion is that of local continuity, or continuity at a point (as opposed to the whole space
		This is version 7 of continuous, born on 2001-10-21, modified 2006-10-22.
		I agree completely with you that the non-standard definition is not shorter or simpler once you take into account the fact that, to use it you first have to define non-standard numbers.
	planetmath.org /encyclopedia/Continuous.html (303 words)

	thms.html
		If a function is continuous on a closed interval, then its extreme values on the interval occur at the endpoints of the interval or at the places interior to the interval where the derivative is 0 or not defined.
		If a function is continuous on a closed interval, differentiable at each point inside the interval, and has the same value at the endpoints of the interval, then its derivative is 0 at some point inside the interval.
		If a function is continuous on a closed interval and differentiable at each point inside the interval, then at some point inside the interval the derivative is equal to the average change in the function over the interval.
	www.ms.uky.edu /~carl/hand98/thms1.html (1257 words)

	s4a4.htm
		In practical work, continuity is important because it means that small errors in the independent variable lead to small errors in the value of the function.
		Note: Requiring a function to be continuous on an interval is not asking very much, as any function whose graph is an unbroken curve over the interval is continuous.
		Rational functions are continuous on any interval in which their denominators are not zero.
	www.csun.edu /~kme52026/s4a4/s4a4.htm (206 words)

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