:* Iterated function systems – use fixed geometric replacement rules ; may be stochastic or deterministic ; e. g., Koch snowflake, Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-Square, Menger sponge

Heine proposed that Cantor solve an open problem that had eluded Dirichlet, Lipschitz, Bernhard Riemann, and Heine himself: the uniqueness of the representation of a function by trigonometric series.

A standard example 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 ).

In set theory, the Cantor – Bernstein – Schroeder theorem, named after Georg Cantor, Felix Bernstein, and Ernst Schröder, states that, if there exist injective functions and between the sets A and B, then there exists a bijective function.

* A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from to to the characteristic function of its graph.

It may happen that a continuous function f is differentiable almost everywhere on, its derivative f ′ is Lebesgue integrable, and nevertheless the integral of f ′ differs from the increment of f. For example, this happens for the Cantor function, which means that this function is not absolutely continuous.

The graph of the Cantor function on the unit interval

In mathematics, the Cantor function, named after Georg Cantor, is an example of a function that is continuous, but not absolutely continuous.

Formally, the Cantor function c: → is defined as follows:

The Cantor function is the most frequently cited example of a real function that is uniformly continuous ( and hence also continuous ) but not absolutely continuous.

Extended to the left with value 0 and to the right with value 1, it is the cumulative probability distribution function of a random variable that is uniformly distributed on the Cantor set.

However, no non-constant part of the Cantor function can be represented as an integral of a probability density function ; integrating any putative probability density function that is not almost everywhere zero over any interval will give positive probability to some interval to which this distribution assigns probability zero.

The Cantor function is the standard example of a singular function.

Cantor said the episode challenges the stereotype that small businesses are public servants who truly care about their customers by portraying local business owner Mr. Tweek as greedier and having fewer scruples than that of the corporation he is challenging ; Tweek knowingly takes advantage of American distrust for big businesses and nostalgia for simpler times in his fight to maintain his bottom line.

However, the term naive set theory is also used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory ; care is required to tell which sense is intended.

In the foundations of mathematics, Russell's paradox ( also known as Russell's antinomy ), discovered by Bertrand Russell in 1901, showed that the naive set theory created by Georg Cantor leads to a contradiction.

In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem.

Implicitly, this is the universe that Georg Cantor was using when he first developed modern naive set theory and cardinality in the 1870s and 1880s in applications to real analysis.

Cantor had apparently discovered the same paradox in his ( Cantor's ) " naive " set theory and this become known as Cantor's paradox.

That the algebra of the real numbers can be employed to yield results about the linear continuum of geometry relies on the Cantor – Dedekind axiom.

In mathematics, the continuum hypothesis ( abbreviated CH ) is a hypothesis, advanced by Georg Cantor in 1878, about the possible sizes of infinite sets.

Before the attacks, Cantor handled about one-quarter of the daily transactions in the multi-trillion dollar treasury security market.

But in 1885, Mittag-Leffler was concerned about the philosophical nature and new terminology in a paper Cantor had submitted to Acta.

He asked Cantor to withdraw the paper from Acta while it was in proof, writing that it was "... about one hundred years too soon.

Cantor attended, hoping to meet Bertrand Russell, whose newly published Principia Mathematica repeatedly cited Cantor's work, but this did not come about.

A 90-minute documentary called loudQUIETloud: a film about The Pixies directed by Steven Cantor and Matthew Galkin was released in 2006.

Two important members of the Notre Dame administration have been suggested as possible identities for Perotinus: the theologian Petrus Cantor ( who died in 1197 ) and the Petrus who was Succentor of Notre Dame from at least 1207 until about 1238.

Fraenkel also was interested in the history of mathematics, writing in 1920 and 1930 about Gauss ' works in algebra, and he published a biography of Georg Cantor.

Mistakes and philosophical weaknesses in reasoning about infinitesimal numbers in the work of Gottfried Leibniz, Johann Bernoulli, Leonhard Euler, Augustin-Louis Cauchy, and others were the reason that they were originally abandoned for the more cumbersome real number-based arguments developed by Georg Cantor, Richard Dedekind, and Karl Weierstrass, which were perceived as being more rigorous by Weierstrass's followers.

Cantor says it is important for " its brilliant summary of European political history from about 1050 to 1350 ".

Also, her association with the paper is noted at the end of any Times-Dispatch story about Rep. Cantor.

The banding evident in the Rössler attractor is similar to a Cantor set rotated about its midpoint.

" Meanwhile, back in New York City, Cantor Rabinovich confronts Rivka about Jess going to California and reminds her that her place is by her husband's side.

A halting probability can be interpreted as the measure of a certain subset of Cantor space under the usual probability measure on Cantor space.

The probability measure on Cantor space, sometimes called the fair-coin measure, is defined so that for any binary string x the set of sequences that begin with x has measure 2 < sup >-| x |</ sup >.

* The Cantor set is an example of an uncountable set that has Lebesgue measure zero.

* Smith – Volterra – Cantor set, a set that is nowhere dense ( in particular it contains no intervals ), yet has positive measure

In spite of this, there exists a homeomorphism of the cube having finite distortion that " squeezes the sponge " in the sense that the holes in the sponge go to a Cantor set of zero measure.

For example, Kahane uses Cantor sets to construct a Besicovitch set of measure zero in the two-dimensional plane.

In mathematics, the Smith – Volterra – Cantor set ( SVC ), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line R that is nowhere dense ( in particular it contains no intervals ), yet has positive measure.

Since the Smith – Volterra – Cantor set S has positive Lebesgue measure, this means that V ′ is discontinuous on a set of positive measure.

* The Smith – Volterra – Cantor set, a Cantor set with measure greater than zero.