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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
Probability distribution
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. This distribution has neither a
Cantor_distribution
Set of points on a line segment with certain topological properties
ISBN 0-19-507068-2. "Cantor set", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Cantor Sets and Cantor Set and Function at cut-the-knot Cantor Set at Platonic
Cantor_set
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
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
Proof in set theory
of this function, which is a subset of R, is uncountable. Therefore, R is uncountable. Also, by using a method of construction devised by Cantor, a bijection
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
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
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
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
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
Curve whose range contains the unit square
Cantor space 𝟐ℕ. We start with a continuous function h from the Cantor space 𝒞 onto the entire unit interval [0,1]. (The restriction of the Cantor function
Space-filling_curve
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
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
Form of continuity for functions
Weierstrass function over any interval is uniformly continuous, but not differentiable anywhere, and hence not of bounded variation; the Cantor function over
Absolute_continuity
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
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
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
Topics referred to by the same term
Topological group Cantor distribution – Probability distribution Cantor function – Continuous function that is not absolutely continuous Cantor Medal – Award
Cantor_(disambiguation)
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
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
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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)
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
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)
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
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
Size of a set in mathematics
Further, the tangent function is a bijection from the interval (−π2,π2) to the whole real line. A more surprising example is the Cantor set, which is defined
Cardinality
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
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
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
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)
American molecular geneticist
Cantor, Charles R.; Schimmel, Paul R. (15 April 1980). Biophysical Chemistry: Part II: Techniques for the Study of Biological Structure and Function.
Charles_Cantor
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
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
Apollonian gasket Attractor Box-counting dimension Cantor distribution Cantor dust Cantor function Cantor set Cantor space Chaos theory Coastline Constructal theory
Index of fractal-related articles
Index_of_fractal-related_articles
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
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
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)
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
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)
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
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
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
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
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
Concept in philosophy and set theory
an extension of the idea of infinity proposed by mathematician Georg Cantor. Cantor linked the absolute infinite with God. Some link the absolute infinite
Absolute_infinite
Structurally stable features in coupled oscillation dynamical systems.
fat Cantor set. Consequently, its fractal dimension is 1. Fixing K and taking a cross-section through the diagram, so that ω is plotted as a function of
Arnold_tongue
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
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
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
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
Type of uniform space
uniformly connected space or Cantor connected space is a uniform space U such that every uniformly continuous function from U to a discrete uniform space
Uniformly_connected_space
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
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
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
Random process of binary (boolean) random variables
y=∑n=0∞bn3n+1 gives the Cantor function, as conventionally defined. This is one reason why the set {H,T}ℕ is sometimes called the Cantor set. Another way to
Bernoulli_process
Counterexample to the converse of the intermediate value theorem
Conway's base 13 function is a mathematical function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate
Conway's_base_13_function
Paradox in set theory
1890s, 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
Mathematical function that can be computed by a program
Computable functions are the basic objects of study in computability theory. Informally, a function is computable if there is an algorithm that computes
Computable_function
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
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
theorems are named after Henri Lebesgue: Blaschke–Lebesgue theorem Cantor–Lebesgue function Borel–Lebesgue theorem Fatou–Lebesgue theorem Lebesgue constant
List of things named after Henri Lebesgue
List_of_things_named_after_Henri_Lebesgue
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
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
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
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
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
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
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
One-to-one correspondence
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the
Bijection
Set with algorithmic membership test
image of A × B under the Cantor pairing function is computable. In general, the image of a computable set under a computable function is computably enumerable
Computable_set
Proposition in mathematical logic
beth numbers: ℶ 1 = ℵ 1 {\displaystyle \beth _{1}=\aleph _{1}} . Georg Cantor advanced the continuum hypothesis in 1878. It became one of the most studied
Continuum_hypothesis
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
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
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)
Concept in mathematical logic and set theory
of Cantor space under the map that takes each function from ω {\displaystyle \omega } to ω {\displaystyle \omega } to the characteristic function of its
Analytical_hierarchy
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
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
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
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
Shape containing unit line segments in all directions
need for Pál joins. There are also other methods; for example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the plane. The Kakeya
Kakeya_set
(sometimes a Hausdorff space) Continuum hypothesis, a conjecture of Georg Cantor that there is no cardinal number between that of countably infinite sets
List of continuity-related mathematical topics
List_of_continuity-related_mathematical_topics
published by German-Russian botanist Wladimir Köppen. Georg Cantor introduces the Cantor function. Gottlob Frege publishes Die Grundlagen der Arithmetik ("The
1884_in_science
Process of repeating items in a self-similar way
defined mathematical objects include factorials, functions (e.g., recurrence relations), sets (e.g., Cantor ternary set), and fractals. There are various
Recursion
Type of mathematical function
schemes by Wilhelm Ackermann, Heinz Bachmann, Wilfried Buchholz, Georg Cantor, Solomon Feferman, Gerhard Jäger, Isles, Pfeiffer, Wolfram Pohlers, Kurt
Ordinal_notation
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
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CANTOR FUNCTION
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