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Topic: Cantor function

Related Topics

Cantor distribution

Cantor set

Almost everywhere

Null set

Cantor space

Lebesgue integral

Derivative

Measure zero

Hausdorff dimension

Peano curve

Fourier series

Category theory

Lebesgue measure

List of real analysis topics

Cantor dust

	PlanetMath: Cantor function
		The Cantor function is a canonical example of a singular function.
		Sometimes this function, and functions similar to it are called the Devil's staircase.
		This is version 5 of Cantor function, born on 2004-02-08, modified 2007-07-10.
	planetmath.org /encyclopedia/CantorFunction.html (286 words)

	Cantor set and function
		When I was a freshman, a graduate student showed me the Cantor set, and remarked that although there were supposed to be points in the set other than the endpoints, he had never been able to find any.
		In other words, a nonconstant function f that has a derivative equal to 0 on intervals of total length 1 manages to grow from 0 to 1 on the interval [0, 1].
		Cantor function is as famous as it is useful for other exceptional constructs.
	www.cut-the-knot.org /do_you_know/cantor.shtml (644 words)

	PlanetMath: Cantor function
		The Cantor function is a canonical example of a singular function.
		Sometimes this function, and functions similar to it are called the Devil's staircase.
		This is version 5 of Cantor function, born on 2004-02-08, modified 2007-07-10.
	www.planetmath.org /encyclopedia/CantorFunction.html (287 words)

	Cantor function - Biocrawler (Site not responding. Last check: )
		This function is the most frequently cited example of a real function that is continuous but not absolutely continuous.
		on the interval that converges to the Cantor function.
		That is, y = y(x) is the Cantor function.
	www.biocrawler.com /encyclopedia/Cantor_function (488 words)

	Cantor set (Site not responding. Last check: )
		Since the Cantor set is the complement of a union of open sets, it itself is a closed subset of the reals, and therefore a complete metric space.
		It is worth noting that as a topological space, the Cantor set is homeomorphic to the product of countably many copies of the space {0, 1}, where each copy carries the discrete topology, as can easily be shown using the binary expansion used to prove its uncountability.
		Cantor himself was led to it by practical concerns about the set of points where a trigonometric series might fail to converge.
	www.xasa.com /wiki/en/wikipedia/c/ca/cantor_set.html (1223 words)

	NationMaster - Encyclopedia: Absolute continuity (Site not responding. Last check: )
		In mathematical analysis, a function f(x) is called uniformly continuous if, roughly speaking, small changes in the input x affect 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...
		The Cantor function is continuous everywhere but not absolutely continuous; as is the function In mathematics, the Cantor function is a function c : [0,1] → [0,1] defined as follows: Express x in base 3.
		In mathematics, given f, a real-valued function on the interval [a, b] on the real line, the total variation of f on that interval is the supremum running over all partitions P = { x1,..., xn } of the interval [a, b].
	www.nationmaster.com /encyclopedia/Absolute-continuity (1048 words)

	Georg Cantor Summary
		Cantor died in a mental hospital in 1918.
		In 1911, Cantor was one of the distinguished foreign scholars invited to attend the 500th anniversary of the founding of the University of St. Andrews in Scotland.
		For Kronecker, Cantor's hierarchy of infinities was inadmissible.
	www.bookrags.com /Georg_Cantor (6336 words)

	Cantor biography
		Cantor was promoted to Extraordinary Professor at Halle in 1872 and in that year he began a friendship with Dedekind who he had met while on holiday in Switzerland.
		Cantor continued to correspond with Dedekind, sharing his ideas and seeking Dedekind's opinions, and he wrote to Dedekind in 1877 proving that there was a 1-1 correspondence of points on the interval [0, 1] and points in p-dimensional space.
		Cantor also discussed the concept of dimension and stressed the fact that his correspondence between the interval [0, 1] and the unit square was not a continuous map.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Cantor.html (3038 words)

	Body
		Cantor sets play an important role in real analysis, particularly in furnishing counter examples and exotic or pathological functions.
		Most frequently, we meet Cantor set of zero measure but the construction is canonical enough to apply to give Cantor set of positive measure.
		We shall next show that g is a non decreasing function, that is g is a monotonic increasing function (not necessarily strictly increasing).
	www.math.nus.edu.sg /~matngtb/Calculus/Cantor/cantor.htm (2458 words)

	Georg Cantor and Cantor's Theorem
		Before Cantor it was generally felt that infinity as an actuality did not make sense; one could only speak of a variable increasing without bound as that variable going to infinity.
		Cantor not only found a way to make sense out an actual, as opposed to a potential, infinity but showed that their are different orders of infinity.
		Cantor first studied at the University of Zürich and later at the University of Berlin.
	www2.sjsu.edu /faculty/watkins/cantorth.htm (514 words)

	Cantor's square w x w table with diagonal
		Now comes the clue of Cantor's diagonal proof: there is no 1-1 map between 2^N and N, resulting in a higher infinity-order of 2^N, called uncountable, versus countable ('linear infinity') N. Fill a square infinite matrix of N rows and N columns with 0's and 1's.
		Cantor's proof of "uncountable" then boils down simply to checking the number of ones, say in the complemented diagonal which is NOT in N obviously because it does not contain just _one_ 1, but none.
		Now Cantor's set 2^N of all subsets of N: here the elements of N are represented for instance in binary code, as strings over {0,1} with the arithmetic interpretation of reals on interval [0,1).
	home.iae.nl /users/benschop/cantor.htm (9036 words)

	A structural point of view on Cantor's diagonal arguments Text - Physics Forums Library
		After we add Cantor's function result to the list, we rearrange it in such a way, which gives us a new rational number as Cantor's function result, and so on and so on.
		But it really doesn't matter, because still we can clime that we have some sequence as Cantor's function result, only because we deal with 2^aleph0-1 sequences, where the missing one is Cantor's function result, and this missing sequence depends on some abitrary order and the rule which we use to define Cantor's function result.
		When we have 2^aleph0 sequences, Cantor's function result is unknown because we have no input at all (no rational form, no base valuse expansion form, no what so ever).
	www.physicsforums.com /archive/index.php/t-7748.html (3311 words)

	[No title]
		The result is returned as a pure function partly for convenience and partly to discourage the user from trying to rewrite it as a linear combination of {1; x, x^2,..., x^n}.
		Last, objects may depend on parameters and become geometrical functions ideal to generate families of objects, this is, the realm of dynamical geometry.
		Functions such as Mean, which give properties of statistical distributions, take the symbolic representation of the distribution as an argument.
	ftp.gwdg.de /pub/math/mathematica/math-source/Listings/Listing.txt (8755 words)

	PlanetMath:
		Cantor function (in Cantor function) owned by jirka
		Cantor's diagonalization (=Cantor's diagonal argument) owned by rmilson
		Cantor ternary function (=Cantor function) owned by jirka
	planetmath.org /encyclopedia/C (4166 words)

	The Milken Archive of American Jewish Music
		Especially in the modern era, this usually implies a "professional" artist as distinguished from a lay precentor, (ba’al t’filla — "master of prayer"), even though the latter must also be fully expert in the intricacies of appropriate and required modes and tunes, but not necessarily musically or vocally trained.
		Bona fide cantors in the modern era are institutionalized clergy with pastoral functions as well, co-equal with rabbis and part of a "dual clergy" unique to Jewish practice.
		The cantor’s function, whether professional or lay, is required for synagogue services.
	www.milkenarchive.org /glossary/glossary.taf?function=detail&ID=6 (129 words)

	[No title]
		Continuing this process indefinitely, you have f defined on the complement of the Cantor set C, where C is an uncountable, nowhere-dense subset of [0,1] having measure zero.
		It is easy to determine that there is a unique extension of f to [0,1] that is continuous on its domain, and that the resulting function has all the properties required.
		In particular, f' is identically zero on the complement of the Cantor set.
	www.math.niu.edu /~rusin/known-math/99/cantor_func (277 words)

	Cantor Set (PRIME)
		This is evidently a function that is surjective.
		We can extend this to a function, called the Cantor function, from the entire unit interval onto itself, by simply agreeing to let its value on the missing intervals be the constant values which equal the values of the original function on the endpoints of those intervals.
		Moreover, every point of the Cantor set is an accumulation point, since within any neighborhood of a number whose ternary expansion consists entirely of 0’s and 2’s one may find other such numbers.
	www.mathacademy.com /pr/prime/articles/cantset (1244 words)

	[No title]
		Cantor set We now extend the idea of Canter function to a two-dimensional object in plane and, in particular, a triangle.
		That is, the Sierpinski pyramid function breaks up the input pyramid into half-size pyramids and throws away all but the four corner ones, as shown in figure 3.
		Cantor gasket Menger sponge Lastly, we apply the Cantor gasket function to a cube in three dimensions, as shown in figure 5.
	mathforum.org /~lisab/lesson5.doc (2230 words)

	[No title]
		For the Cantor set above, if e is 3 times smaller we need 2 times larger N (because holes do not need to be covered), thus D = log2 / log3 = 0.63....
		Non-integer dimension is a typical property of fractals and fractal structure is typical for strange chaotic attractors.
		CA evolves according to a deterministic rule which gives the state of a site in the next generation as a function of the states of neighboring sites in the present generation.
	www.vscht.cz /mat/Pavel.Pokorny/nlnfaq.draft (1432 words)

	[No title]
		Give an example of a function that is Lebesgue integrable, but not Riemann integrable.
		Prove that a sequence of continuous functions on [0,1] that converges monotonically to 0 converges uniformly (Dini's Thm).
		Suppose you have a sequence of functions f_n of total variation that is bounded independent of n.
	www.princeton.edu /~missouri/Generals/generals/realan.txt (976 words)

	10.5. Cantor, Georg (1845-1918)
		It was in that same year of 1874 that Cantor published his first paper on the theory of sets.
		One great mathematician, Henri Poincare expressed his disapproval, stating that Cantor's set theory would be considered by future generations as "a disease from which one has recovered." However, he was kinder than another critic, Leopold Kronecker.
		Among other things, he delayed or suppressed completely Cantor's and his followers' publications, raged both written and verbal personal attacks against him, belittled his ideas in front of his students and blocked Cantor's life ambition of gaining a position at the prestigious University of Berlin.
	web01.shu.edu /projects/reals/history/cantor.html (967 words)

	Cantor Function
		The Cantor function is a function that is continuous, differentiable, increasing, non-constant, and the derivative is zero everywhere except at a set with length zero.
		This is the most difficult function in our repertoire and can be found, for example, in Kolmogorov and Fomin.
		That is a crucial point, because we used this limit to extend the function to the whole interval [0, 1].
	web01.shu.edu /projects/reals/cont/fp_cantr.html (300 words)

	Leipzig 2
		The cantor lived in the left (southern) part of the school, the rector in the right (northern) part, and the school boys in the middle.
		Traditionally, this had been the privilege of the cantor, and when Ernesti tried to take over the cantor's role in this respect, Bach was furious (the enlightened Ernesti, one of the smartest faces in our portrait gallery in my opinion, wanted to dismiss a prefect that had beaten a student; this was against Bach's will).
		Bach had applied for this function in 1733, with a very humble letter and under submission of two parts (the Kyrie and Gloria) of the (later) Mass in B Minor.
	odur.let.rug.nl /Linguistics/diversen/bach/leipzig2.html (1531 words)

	Math Forum Discussions
		The Cantor function is continuous, so of course it's Riemann integrable.
		The derivative of the Cantor function is not everywhere defined, and
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/message.jspa?messageID=3681194&tstart=0 (155 words)

	Cantor's Paradox
		Cantor's paradox, sometimes called the paradox of the greatest cardinal, expresses what its second name would imply--that there is no cardinal larger than every other cardinal.
		Cantor proves his theorems in two ways, function-theoretically and by using a diagonal argument.
		These conclusions are sound, given that Cantor stated in his letter that the ``totality of everything thinkable'' ([11], p.
	www.u.arizona.edu /~miller/finalreport/node3.html (444 words)

	cantor - definition by dict.die.net
		cantor n 1: the musical director of a choir [syn: choirmaster, precentor] 2: the official of a synagogue who conducts the liturgical part of the service and sings or chants the prayers intended to be performed as solos [syn: hazan]
		Cantor devised the diagonal proof of the uncountability of the real numbers: Given a function, f, from the natural numbers to the real numbers, consider the real number r whose binary expansion is given as follows: for each natural number i, r's i-th digit is the complement of the i-th digit of f(i).
		This is just a special case of a diagonal proof that a function from a set to its power set cannot be surjective: Let f be a function from a set S to its power set, P(S) and let U = x in S: x not in f(x).
	dict.die.net /cantor (301 words)

	NationMaster - Encyclopedia: Cantor distribution (Site not responding. Last check: )
		The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
		This distribution is not absolutely continuous with respect to Lebesgue measure, so it has no probability density function; neither is it discrete, since it has no point-masses; nor is it even a mixture of a discrete probability distribution with one that has a density function.
		Then the probability distribution of X is the Cantor distribution.
	www.nationmaster.com /encyclopedia/Cantor-distribution (450 words)

	Cantor Fitzgerald
		Cantor Fitzgerald has been working with the DOJ to advocate policies that it believes will help the families get all the compensation they deserve as soon as possible.
		These voluntary payments include Cantor Fitzgerald L.P.’s decision to distribute for the benefit of the families an amount equal to 25% of the partnership's profits that otherwise would have been distributed to the remaining partners for the next 5 years, as well as charitable monies distributed by the Cantor Fitzgerald Relief Fund.
		Cantor Fitzgerald has pledged to benefit the families of lost Cantor, eSpeed and TradeSpark employees in an amount equal to 25% of the profits, for the next five years that would otherwise be distributed to partners.
	cantorrelief.org /pages/1713172 (4062 words)

	Popular Culture and Spontaneous Order or How I Learned To Stop Worrying and Love the Tube by Paul A. Cantor
		According to Romantic aesthetics, in a poem every word has an artistic function to perform in the work as a whole, but in a novel, many words are there simply because the novelist was being paid by the word.
		Observing the similarities in the way serial production functioned in the nineteenth-century novel and in twentieth-century television is a good way of seeing how much high culture and popular culture have in common.
		Paul A. Cantor [send him mail] is Professor of English at the University of Virginia and author of Gilligan Unbound: Pop Culture in the Age of Globalization.
	www.lewrockwell.com /orig3/cantor4.html (10436 words)

	Re: Cantor's set sizes
		Cantor was led to these investigations because he was interested in >conditions where the Fourier series representation of a function actually >represents the function.
		Cantor showed that if a trig series converges to 0 except at finitely many points, all the coefficients are 0.
		Cantor tried to show that countble sets are sets of uniqueness, but failed, even though the result is true.
	www.usenet.com /newsgroups/sci.math/msg07524.html (209 words)

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