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Continuous function that is not absolutely continuous
In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous. It is a notorious counterexample in
Cantor_function
Function uniquely mapping two numbers into a single number
pairing function can also be generalized: there exists an n-ary generalized Cantor pairing function on N {\displaystyle \mathbb {N} } . The Cantor pairing
Pairing_function
Set of points on a line segment with certain topological properties
In mathematics, the Cantor set is a self-similar set of points lying on a single line segment that has a number of unintuitive properties. It was discovered
Cantor_set
Probability distribution
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a
Cantor_distribution
Proof in set theory
Cantor's diagonal argument (among various similar names) is a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence
Cantor's_diagonal_argument
Mathematician (1845–1918)
Georg Ferdinand Ludwig Philipp Cantor (/ˈkæntɔːr/ KAN-tor; German: [ˈɡeːɔʁk ˈfɛʁdinant ˈluːtvɪç ˈfɪlɪp ˈkantoːɐ̯]; 3 March [O.S. 19 February] 1845 – 6
Georg_Cantor
Topological space
mathematics, a Cantor space, named for Georg Cantor, is a topological abstraction of the classical Cantor set: a topological space is a Cantor space if it
Cantor_space
Theorem in set theory
Bernstein. It is also known as the Cantor–Bernstein theorem or Cantor–Schröder–Bernstein theorem, after Georg Cantor, who first published it (albeit without
Schröder–Bernstein_theorem
In mathematics, with negligible exceptions
normal. The Cantor set is also null. Thus, almost all reals are not in it even though it is uncountable. The derivative of the Cantor function is 0 for almost
Almost_all
Curve whose range contains the unit square
(The restriction of the Cantor function to the Cantor set is an example of such a function.) From it, we get a continuous function H {\displaystyle H} from
Space-filling_curve
Order-preserving mathematical function
see Cantor function. if this set is countable, then f {\displaystyle f} is absolutely continuous if f {\displaystyle f} is a monotonic function defined
Monotonic_function
Type of function
of a singular function is the Cantor function, which is sometimes called the devil's staircase (a term also used for singular functions in general). There
Singular_function
Differentiable function whose derivative is not Riemann integrable
Riemann-integrable. The function is defined by making use of the Smith–Volterra–Cantor set and an infinite number or "copies" of sections of the function defined by
Volterra's_function
Every set is smaller than its power set
X} to Y {\displaystyle Y} . This is the heart of Cantor's theorem: there is no surjective function from any set A {\displaystyle A} to its power set
Cantor's_theorem
Continuous fractal curve obtained as the image of Cantor space
well-known fractal curves, including the Cantor function, Cesàro–Faber curve (Lévy C curve), Minkowski's question mark function, blancmange curve, and the Koch
De_Rham_curve
Form of continuity for functions
example with the Cantor function. Let I {\displaystyle I} be an interval in the real line R {\displaystyle \mathbb {R} } . A function f : I → R {\displaystyle
Absolute_continuity
Uniform restraint of the change in functions
Any absolutely continuous function (over a compact interval) is uniformly continuous. On the other hand, the Cantor function is uniformly continuous but
Uniform_continuity
Topics referred to by the same term
Cantor distribution Cantor function Cantor medal, German mathematics prize named after Georg Cantor Cantor set Cantor space Cantor's theorem (disambiguation)
Cantor_(disambiguation)
Generalization of the Riemann integral
cumulative distribution function g is continuous, it does not work if g fails to be absolutely continuous (again, the Cantor function may serve as an example
Riemann–Stieltjes_integral
Counterintuitive mathematical object
Dirichlet function, which is the indicator function for rationals, is a bounded function that is not Riemann integrable. The Cantor function is a monotonic
Pathological_(mathematics)
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function shows that f A {\displaystyle f_{A}} has A as its set of discontinuities. Blumberg theorem Cantor function Dirichlet function Euclid's
Thomae's_function
Set of real numbers in mathematics
In mathematics, the Smith–Volterra–Cantor set (SVC), ε-Cantor set, or fat Cantor set is an example of a set of points on the real line that is nowhere
Smith–Volterra–Cantor_set
Type of continuity of a complex-valued function
continuous by the Heine-Cantor theorem. It does not satisfy a Hölder condition of any order, however. The Weierstrass function defined by: f ( x ) = ∑
Hölder_condition
Function with unusual fractal properties
147. Cantor function, which can be understood as reinterpreting ternary numbers as binary numbers, analogously to the way the question-mark function reinterprets
Minkowski's question-mark function
Minkowski's_question-mark_function
Relationship between derivatives and integrals
may fail for continuous functions F that admit a derivative f(x) at almost every point x, as the example of the Cantor function shows. However, if F is
Fundamental theorem of calculus
Fundamental_theorem_of_calculus
Everywhere except a set of measure zero
almost all of the factors. Dirichlet's function, a function that is equal to 0 almost everywhere. Cantor function Weisstein, Eric W. "Almost Everywhere"
Almost_everywhere
Measurable set whose measure is zero
a nonmeasurable subset. Let f {\displaystyle f} be the Cantor function, a continuous function which is locally constant on K c , {\displaystyle K^{c}
Null_set
Topics referred to by the same term
by Santa Clara Vanguard Drum and Bugle Corps a singular function in mathematics Cantor function Baguenaudier, a disentanglement puzzle This disambiguation
Devil's_staircase
Musical director of the Thomanerchor in Leipzig
the Thomaskantor in Latin, Cantor et Director Musices, describes the two functions of cantor and director. As the cantor, he prepared the choir for service
Thomaskantor
Cantor (1845–1918), a German mathematician. Cantor algebra Cantor cube Cantor distribution Cantor function Cantor normal form Cantor pairing function
List of things named after Georg Cantor
List_of_things_named_after_Georg_Cantor
Real function with finite total variation
V_{a}^{b}(f)=|f(b)-f(a)|.} In particular, the monotone Cantor function is a well-known example of a function of bounded variation that is not absolutely continuous
Bounded_variation
About mathematical infinity
mathematical logic, the theory of infinite sets was first developed by Georg Cantor. Although this work has become a thoroughly standard fixture of classical
Controversy over Cantor's theory
Controversy_over_Cantor's_theory
Fractal curve resembling a blancmange pudding
the blancmange curve. Cantor function (also known as the Devil's staircase) Minkowski's question mark function Weierstrass function Dyadic transformation
Blancmange_curve
Mathematical theorem
In mathematics, the Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact.
Heine–Cantor_theorem
First article on transfinite set theory
Cantor's first set theory article contains Georg Cantor's first theorems of transfinite set theory, which studies infinite sets and their properties.
Cantor's first set theory article
Cantor's_first_set_theory_article
Constant expressing ambiguity from indefinite integrals
theorem still fails. As an example, take F {\displaystyle F} to be the Cantor function and again let G = 0. {\displaystyle G=0.} It turns out that adding
Constant_of_integration
Generalisation of the derivative of a function
{\displaystyle 1_{\mathbb {Q} }} is identified with the zero function. The Cantor function c does not have a weak derivative, despite being differentiable
Weak_derivative
Extremely small quantity in calculus; thing so small that there is no way to measure it
Cours d'Analyse, and in defining an early form of a Dirac delta function. As Cantor and Dedekind were developing more abstract versions of Stevin's continuum
Infinitesimal
Theorem in mathematical measure theory
of measures. The Cantor measure (the probability measure on the real line whose cumulative distribution function is the Cantor function) is an example of
Lebesgue's decomposition theorem
Lebesgue's_decomposition_theorem
Alternate way to define a function in APL
0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 1 0 1 Cantor 0 to Cantor 6 depicted as black bars: The function sieve ⍵ computes a bit vector of length ⍵ so that
Direct_function
Software for scientific statistics and analysis
assistant dialogs for common tasks (like integrating a function or entering a matrix) Cantor was the first KDE project to implement upload to the GetHotNewStuff
Cantor_(mathematics_software)
Set of études by György Ligeti
continuous function in mathematics, known as the Devil's Staircase or Cantor function. The structure of the piece adheres to the properties of the function both
Études_(Ligeti)
British mathematician (1826–1883)
Cantor Set and Cantor Function". Mathematics Magazine. 67 (2): 136–140. doi:10.1080/0025570X.1994.11996201 – via Taylor and Francis+NEJM. The Cantor Set
Henry_John_Stephen_Smith
subject. Alexander horned sphere All horses are the same color Cantor function Cantor set Checking if a coin is biased Concrete illustration of the central
List_of_mathematical_examples
Property of uniformly space-filling movement
}=2^{\mathbb {N} }.} This set is the Cantor set, sometimes called the Cantor space to avoid confusion with the Cantor function C ( x ) = ∑ n = 1 ∞ x n 3 n .
Ergodicity
Chief singer employed at a church
In Christianity, the cantor, female chantress, sometimes called the precentor or the protopsaltes (Greek: πρωτοψάλτης, lit. 'first singer'; from Greek:
Cantor_(Christianity)
Size of a set in mathematics
first introduced formally to mathematics by Georg Cantor at the turn of the 20th century. Cantor's theory of cardinality was then formalized, popularized
Cardinality
Collection of mathematical objects
may be located. The mathematical study of infinite sets began with Georg Cantor (1845–1918). This provided some counterintuitive statements and paradoxes
Set_(mathematics)
Mathematical set that can be enumerated
no surjective function from A {\displaystyle A} to P ( A ) {\displaystyle {\mathcal {P}}(A)} . A proof is given in the article Cantor's theorem. As an
Countable_set
Description of continuous random distribution
probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has
Probability_density_function
Paradox in set theory
Georg Cantor – considered the founder of modern set theory – had already realized that his theory would lead to a contradiction (to Cantor's theorem)
Russell's_paradox
Doubling map on the unit interval
{\displaystyle y=\sum _{n=0}^{\infty }{\frac {b_{n}}{3^{n+1}}}} gives the Cantor function, as conventionally defined. This is one reason why the set { H , T
Dyadic_transformation
Branch of mathematics that studies sets
in real analysis of the study of “seriously” discontinuous functions. A young Georg Cantor entered into this area, which led him to the study of point-sets
Set_theory
Fractal sets in complex dynamics of mathematics
(Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby
Julia_set
Size of a possibly infinite set
as follows: |X| ≤ |Y| means that there exists an injective function from X to Y. The Cantor–Bernstein–Schroeder theorem states that if |X| ≤ |Y| and |Y|
Cardinal_number
Unique strong solution of a stochastic differential equation
with respect to time; for example, X {\displaystyle X} can be the Cantor function. A stochastic exponential cannot go to zero continuously; it can only
Doléans-Dade_exponential
ambo and its function was described by the oldest Ordo, Ordo romanus, as follows: "After the subdeacon has read the Epistle, the cantor ascends to the
Cantatorium
American molecular geneticist
Charles R. Cantor (born August 26, 1942 in Brooklyn) is an American molecular geneticist who, in conjunction with David Schwartz, developed pulse field
Charles_Cantor
Infinite cardinal number
size) of infinite sets. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter
Aleph_number
Operations on ordinals that extend classical arithmetic
As discussed above, the Cantor normal form of ordinals below ε0 can be expressed in an alphabet containing only the function symbols for addition, multiplication
Ordinal_arithmetic
Halting probability of a random computer program
interpreted as the measure of a certain subset of Cantor space under the usual probability measure on Cantor space. It is from this interpretation that halting
Chaitin's_constant
Vector space of functions in mathematics
integral of its derivative (this excludes irrelevant examples such as Cantor's function). With this definition, the Sobolev spaces admit a natural norm, ‖
Sobolev_space
Random process of binary (boolean) random variables
{\displaystyle y=\sum _{n=0}^{\infty }{\frac {b_{n}}{3^{n+1}}}} gives the Cantor function, as conventionally defined. This is one reason why the set { H , T
Bernoulli_process
List of terms created from a person's name
reaction Georg Cantor, German mathematician – Cantor algebra, Cantor cube, Cantor function, Cantor space, Cantor's back-and-forth method, Cantor–Bernstein
List_of_eponyms_(A–K)
Orientation-preserving mapping class group of the torus
describes the self-similarity symmetries of the Cantor function, Minkowski's question mark function, and the Koch snowflake, each being a special case
Modular_group
Phenomenon in maths
that ω is plotted as a function of Ω, gives the "Devil's staircase", a shape that is generically similar to the Cantor function. One can show that for
Arnold_tongue
numbers.) The characteristic function of the Cantor set, which equals 1 if x is in the Cantor set and 0 otherwise. This function is 0 for an uncountable set
Baire_function
About mathematical functions
the invention of set theory by Georg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set
History of the function concept
History_of_the_function_concept
Infinite set that is not countable
{\displaystyle \beth _{1}} (beth-one). The Cantor set is an uncountable subset of R {\displaystyle \mathbb {R} } . The Cantor set is a fractal and has Hausdorff
Uncountable_set
Topics referred to by the same term
a non-empty intersection Heine–Cantor theorem: a continuous function on a compact space is uniformly continuous Cantor–Bendixson theorem: a closed set
Cantor's theorem (disambiguation)
Cantor's_theorem_(disambiguation)
Paradox in set theory
In set theory, Cantor's paradox states that there is no set of all cardinalities. This is derived from the theorem that there is no greatest cardinal number
Cantor's_paradox
Measure theory concept
not to functions differentiable on a conull set: The Cantor function does not have the Luzin N property, as the Lebesgue measure of the Cantor set is
Luzin_N_property
Mathematical concept
tangent function, which provides a one-to-one correspondence between the interval (−π/2, π/2) and R. The second result was proved by Cantor in 1878
Infinity
Function that preserves distinctness
In mathematics, an injective function (also known as injection, or one-to-one function) is a function f that maps distinct elements of its domain to distinct
Injective_function
Definition of mathematical integration
approximate derivative is merely of Lebesgue measure zero, as the Cantor function shows. (Gordon 1994, theorem 4.12) (Gordon 1994, theorem 4.14) (Bruckner
Khinchin_integral
Uniqueness of countable dense linear orders
In order theory and model theory, branches of mathematics, Cantor's isomorphism theorem states that every two nonempty countable dense unbounded linear
Cantor's_isomorphism_theorem
Mathematical set containing no elements
example, Cantor defined two sets as being disjoint if their intersection has an absence of points; however, it is debatable whether Cantor viewed O {\displaystyle
Empty_set
Set of all things that may be the input of a mathematical function
In mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom ( f ) {\displaystyle \operatorname
Domain_of_a_function
Hierarchy of complexity classes for formulas defining sets
of Cantor space under the map that takes each function from ω {\displaystyle \omega } to ω {\displaystyle \omega } to the characteristic function of its
Arithmetical_hierarchy
Mathematical function such that every output has at least one input
surjective function (also known as surjection, or onto function /ˈɒn.tuː/) is a function f such that, for every element y of the function's codomain, there
Surjective_function
Proposition in mathematical logic
\beth _{1}=\aleph _{1}} . The continuum hypothesis was advanced by Georg Cantor in 1878. It became one of the most studied problems in set theory, and establishing
Continuum_hypothesis
Open subset of the real–number line
our example, the boundary Cantor string is the Cantor set itself. So the abscissa of convergence of the geometric zeta function ζ L ( s ) {\displaystyle
Fractal_string
The only quadratic pairing functions are the Cantor polynomials
quadratic polynomial pairing functions are the Cantor polynomials. In 1873, Georg Cantor showed that the so-called Cantor polynomial P ( x , y ) := 1 2
Fueter–Pólya_theorem
Mathematical-logic system based on functions
as λ-calculus) is a formal system for expressing computation based on function abstraction and application using variable binding and substitution. Untyped
Lambda_calculus
Branch of mathematics
infinitesimals, and thus, at last, made it logically secure. Next came Georg Cantor, who developed the theory of continuity and infinite numbers. "Continuity"
Calculus
American immunologist
Harvey Cantor is an American immunologist known for his studies of the development and immunological function of T lymphocytes. Cantor is currently the
Harvey_Cantor
Well-quasi-ordering of finite trees
application of the theorem gives the existence of a fast-growing TREE function. TREE(3) is one of the largest simply defined finite numbers, dwarfing
Kruskal's_tree_theorem
Branch of mathematics
application to other fields of mathematics. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli
Topology
Generalization of "n-th" to infinite cases
= ∅. Cantor's work with derived sets and ordinal numbers led to the Cantor-Bendixson theorem. Using successors, limits, and cardinality, Cantor generated
Ordinal_number
German mathematician (1831–1916)
49 64 81 100 ... Dedekind's work in this area anticipated that of Georg Cantor, who is commonly considered the founder of set theory. Likewise, his contributions
Richard_Dedekind
Informal set theories
determining this with certainty is that Cantor did not provide an axiomatization of his system. By 1899, Cantor was aware of some of the paradoxes following
Naive_set_theory
Three-dimensional fractal
concept of topological dimension. It has similar properties as the Cantor set and the Cantor dust, because the construction requires in both cases the removal
Menger_sponge
Function returning one of only two values
switching function, used especially in older computer science literature, and truth function (or logical function), used in logic. Boolean functions are the
Boolean_function
Algorithm for factoring polynomials over finite fields
In computational algebra, the Cantor–Zassenhaus algorithm is a method for factoring polynomials over finite fields (also called Galois fields). The algorithm
Cantor–Zassenhaus_algorithm
Function in Boolean algebra
that all subsets of the Cantor space are measurable and have the property of Baire and thus that no infinite parity function exists; this holds in the
Parity_function
Term in stochastic calculus
respect to time; for example, Y {\displaystyle Y} can equal 1 plus the Cantor function. Stochastic logarithm is an inverse operation to stochastic exponential:
Stochastic_logarithm
Type of transfinite numbers
and multiplication. The original epsilon numbers were introduced by Georg Cantor in the context of ordinal arithmetic; they are the ordinal numbers ε that
Epsilon_number
Cardinality of the set of real numbers
Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between
Cardinality_of_the_continuum
published by German-Russian botanist Wladimir Köppen. Georg Cantor introduces the Cantor function. Gottlob Frege publishes Die Grundlagen der Arithmetik ("The
1884_in_science
American financial services company
as part of the larger Cantor Fitzgerald organization, BGC Partners became its own entity in 2004. In 1945, Bernard Gerald Cantor founded a brokerage service
BGC_Group
CANTOR FUNCTION
CANTOR FUNCTION
Male
Spanish
Spanish name derived from Latin Pastor, PASTOR means "shepherd." St. Pastor was a 9-year-old boy who along with his 13-year-old brother, Justus, was martyred at Alcalá de Henares in the early 4th century.
Surname or Lastname
English
English : variant spelling of Canter.German and Jewish (Ashkenazic) : variant spelling of Kantor.French (Picardy) : learned form of chantre ‘singer’. Compare Canter 1.
Surname or Lastname
English
English : probably a variant of Mander.Belcher Manter is recorded in Plymouth, MA, in 1657. John Manter (1658–1744), possibly a son of Belcher, was the founder of a family associated with Martha’s Vineyard.
Girl/Female
Arabic, Muslim
Small Bridge
Male
Greek
(ÎœÎντωÏ) Greek name derived from the word menos, MENTOR means "spirit." In mythology, this is the name of the son of Ãlkimos.
Male
Norwegian
 Norwegian form of Old Norse Arnþórr, ANDOR means "eagle of Thor." Compare with another form of Andor.
Boy/Male
Greek Latin
Beaver. Brother of Helen.
Boy/Male
Danish, French, German, Greek, Latin, Swedish
Brother of Helen; Braver
Surname or Lastname
English
English : from an agent derivative of Anglo-Norman French cant ‘song’, applied as an occupational name for a singer in a chantry or a nickname for someone who had a good voice or who sang a lot.Americanized spelling of Kanter or Kantor.
Surname or Lastname
English
English : habitational name from a place in Norfolk named Caston, from an unattested Old English personal name Catt or the Old Norse personal name Káti + Old English tūn ‘farmstead’, ‘settlement’.
Male
Spanish
Portuguese and Spanish name SANTOS means "saints."Â This name is sometimes bestowed on a child to invoke the protection of the saints. It is also given to baby boys born on the Feast of All Saints.
Surname or Lastname
English (mainly Cambridgeshire)
English (mainly Cambridgeshire) : habitational name from a place in Lincolnshire called Panton, from Old English pamp ‘hill’, ‘ridge’ or panne ‘pan’ + tūn ‘enclosure’, ‘settlement’.
Male
English
Anglicized form of Irish Conchobhar, CONNOR means "hound-lover."
Boy/Male
Latin
Singer.
Surname or Lastname
English
English : habitational name from places called Caistor, in Lincolnshire and Norfolk, Caister in Norfolk, or Castor in Cambridgeshire, all named with Old English cæster ‘Roman fort or town’.
Boy/Male
American, Australian, British, Chinese, Christian, Danish, English, German, Indian
Transporter of Goods with a Cart; Cart Driver; Carter; Someone who Uses a Cart
Surname or Lastname
French and Italian
French and Italian : occupational name from French, northern Italian sartor ‘tailor’ (Latin sartor).English : topographic name denoting someone who lived on land which had been cleared for cultivation, Old French assart, essart ‘woodland cleared for cultivation’ + the habitational suffix -er.
Male
Hungarian
 Variant spelling of Hungarian András, ANDOR means "man; warrior." Compare with another form of Andor.
Male
Greek
(ΚάστωÏ) Greek name KASTOR means "beaver." In mythology, Castor/Kastor and Pollux/Polydeukes ("very sweet") are the twin sons of Leda and are known as the Gemini twins.
Male
English
English occupational surname transferred to forename use, CARTER means "carter," someone who uses a cart.
CANTOR FUNCTION
CANTOR FUNCTION
Female
Egyptian
, the wife of King Namrut.
Girl/Female
Australian, Hindu, Indian
Dancing Stars
Female
Chinese
flowery tuber.
Girl/Female
Irish
Golden.
Girl/Female
Muslim
Beloved
Girl/Female
Australian, Greek
Flower Name
Female
English
Feminine form of English John, JONIE means "God is gracious."
Girl/Female
Tamil
Kankangi | கநà¯à®•à®¾à®¨à®•à¯€
Gold
Boy/Male
Muslim
Noble, Generous
Girl/Female
Arabic, Indian, Pakistani
Combination
CANTOR FUNCTION
CANTOR FUNCTION
CANTOR FUNCTION
CANTOR FUNCTION
CANTOR FUNCTION
a.
Of or pertaining to a cantor; as, the cantoris side of a choir; a cantoris stall.
n.
A kind of type. See Canon.
a.
Of or pertaining to a canton or cantons; of the nature of a canton.
pl.
of Cento
a.
Of or belonging to a cantor.
n.
One who cants or whines; a beggar.
n.
See Caster, a small wheel.
a.
Having angles; as, a six canted bolt head; a canted window.
pl.
of Canto
n.
See Cantle.
n.
See Center.
pl.
of Cannon
v. t.
To cause, as a horse, to go at a canter; to ride (a horse) at a canter.
n.
A song or canto
imp. & p. p.
of Cant
v. i.
To move in a canter.
v. i.
The canto, cantus, or soprano voice; the treble.
n.
A chanter.
n.
One who casts; as, caster of stones, etc. ; a caster of cannon; a caster of accounts.
a.
Eaten out by canker, or as by canker.