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Nowhere Continuous Function
In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f is a function from real numbers to real numbers, then f is nowhere continuous if for each point x there is some \varepsilon > 0 such that for every \delta > 0, we can find a point y such that , x - y, < \delta and . Therefore, no matter how close it gets to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the by the distance function in a , or by using the definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dense Set
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation). Formally, A is dense in X if the smallest closed subset of X containing A is X itself. The of a topological space X is the least cardinality of a dense subset of X. Definition A subset A of a topological space X is said to be a of X if any of the following equivalent conditions are satisfied: The smallest closed subset of X containing A is X itself. The closure of A in X is equal to X. That is, \operatorname_X A = X. The interior of the complement of A is empty. That is, \operatorname_X (X \setminus A) = \varnothing. Every point in X eith ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Parents' Choice Award in 2008. Wolfram Research's staff organizes and edits the Demonstrations, which may be created by any user of Mathematica, then freely published and freely downloaded. Technology The Demonstrations run in Mathematica 6 or above and in Wolfram CDF Player, which is a free modified version of Wolfram Mathematica and available for Windows, Linux, and macOS and can operate as a web browser plugin. Demonstrations can also be embedded into a website. Each Demonstration page includes a snippet of JavaScript JavaScript (), often abbreviated as JS, is a programming language and core technology of the World Wide Web, alongside HTML and CSS. Ninety-nine percent of websites use JavaScript on the client side for webp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Function
In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function (mathematics), function that is continuous function, continuous everywhere but Differentiable function, differentiable nowhere. It is also an example of a fractal curve. The Weierstrass function has historically served the role of a pathological (mathematics), pathological function, being the first published example (1872) specifically concocted to challenge the notion that every continuous function is differentiable except on a set of isolated points. Weierstrass's demonstration that continuity did not imply almost-everywhere differentiability upended mathematics, overturning several proofs that relied on geometric intuition and vague definitions of smoothness. These types of functions were disliked by contemporaries: Charles Hermite, on finding that one class of function he was working on had such a property, described it as a "lamentable scourge". Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thomae's Function
Thomae's function is a real-valued function of a real variable that can be defined as: f(x) = \begin \frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\ 0 &\textx \text \end It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), the Riemann function, or the Stars over Babylon (John Horton Conway's name). Thomae mentioned it as an example for an integrable function with infinitely many discontinuities in an early textbook on Riemann's notion of integration. Since every rational number has a unique representation with coprime (also termed relatively prime) p \in \mathbb Z and q \in \mathbb N, the function is well-defined. Note that q = +1 is the only number in \mathbb N that is coprime to p = 0. It is a modification of the Dirichlet function, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Blumberg Theorem
In mathematics, the Blumberg theorem states that for any real function f : \Reals \to \Reals there is a Dense set, dense subset D of \Reals such that the Restriction (mathematics), restriction of f to D is continuous function, continuous. It is named after its discoverer, the Russian-American mathematician Henry Blumberg. Examples For instance, the restriction of the Dirichlet function (the indicator function of the rational numbers \Q) to \Q is continuous, although the Dirichlet function is Nowhere continuous function, nowhere continuous in \Reals. Blumberg spaces More generally, a Blumberg space is a topological space X for which any function f : X \to \Reals admits a continuous restriction on a dense subset of X. The Blumberg theorem therefore asserts that \mathbb (equipped with its usual topology) is a Blumberg space. If X is a metric space then X is a Blumberg space if and only if it is a Baire space. The Blumberg problem is to determine whether a compact Hausdorff space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Infinitesimal
In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referred to the "infinity- th" item in a sequence. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. This definition was not rigorously formalized. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. In the 3rd century BC Archimedes used what ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperreal Number
In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conway Base 13 Function
The Conway base 13 function is a function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem. In other words, it is a function that satisfies a particular intermediate-value property — on any interval (a,b), the function f takes every value between f(a) and f(b) — but is not continuous. In 2018, a much simpler function with the property that every open set is mapped onto the full real line, was published by user Aksel Bergfeldt on the community question and answer site Mathematics Stack Exchange. This function is also nowhere continuous. Purpose The Conway base 13 function was created as part of a "produce" activity: in this case, the challenge was to produce a simple-to-understand function which takes on every real value in every interval, that is, it is an everywhere surjective function. It is thus discontinuous at every point. Sketch of definition * Every real number x can be represented in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Discontinuous Linear Functional
In mathematics, linear maps form an important class of "simple" functions which preserve the algebraic structure of linear spaces and are often used as approximations to more general functions (see linear approximation). If the spaces involved are also topological spaces (that is, topological vector spaces), then it makes sense to ask whether all linear maps are continuous. It turns out that for maps defined on infinite- dimensional topological vector spaces (e.g., infinite-dimensional normed spaces), the answer is generally no: there exist discontinuous linear maps. If the domain of definition is complete, it is trickier; such maps can be proven to exist, but the proof relies on the axiom of choice and does not provide an explicit example. A linear map from a finite-dimensional space is always continuous Let ''X'' and ''Y'' be two normed spaces and f : X \to Y a linear map from ''X'' to ''Y''. If ''X'' is finite-dimensional, choose a basis \left(e_1, e_2, \ldots, e_n\right) in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Functional
In mathematics, a linear form (also known as a linear functional, a one-form, or a covector) is a linear mapIn some texts the roles are reversed and vectors are defined as linear maps from covectors to scalars from a vector space to its field of scalars (often, the real numbers or the complex numbers). If is a vector space over a field , the set of all linear functionals from to is itself a vector space over with addition and scalar multiplication defined pointwise. This space is called the dual space of , or sometimes the algebraic dual space, when a topological dual space is also considered. It is often denoted , p. 19, ยง3.1 or, when the field is understood, V^*; other notations are also used, such as V', V^ or V^. When vectors are represented by column vectors (as is common when a basis is fixed), then linear functionals are represented as row vectors, and their values on specific vectors are given by matrix products (with the row vector on the left). Examples T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
