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Denjoy–Young–Saks Theorem
In mathematics, the Denjoy–Young–Saks theorem gives some possibilities for the Dini derivatives of a function that hold almost everywhere. proved the theorem for continuous functions, extended it to measurable function In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...s, and extended it to arbitrary functions. and give historical accounts of the theorem. Statement If ''f'' is a real valued function defined on an interval, then with the possible exception of a set of measure 0 on the interval, the Dini derivatives of ''f'' satisfy one of the following four conditions at each point: *''f'' has a finite derivative *''D''+''f'' = ''D''–''f'' is finite, ''D''−''f'' = ∞, ''D''+''f'' = –∞. *''D''−''f'' = ''D''+''f'' is finite, ''D''+''f'' = ∞, ''D''–''f'' = –� ... [...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] |
Dini Derivative
In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini derivative, which is also called an upper right-hand derivative, of a continuous function :f: \rightarrow , is denoted by and defined by :f'_+(t) = \limsup_ \frac, where is the supremum limit and the limit is a one-sided limit. The lower Dini derivative, , is defined by :f'_-(t) = \liminf_ \frac, where is the infimum limit. If is defined on a vector space, then the upper Dini derivative at in the direction is defined by :f'_+ (t,d) = \limsup_ \frac. If is locally Lipschitz, then is finite. If is differentiable at , then the Dini derivative at is the usual derivative at . Remarks * The functions are defined in terms of the infimum and supremum in order to make the Dini derivatives as "bullet proof" as possible, so tha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of '' almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complement ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Function
In mathematics, a continuous function is a function such that a small variation of the argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as '' discontinuities''. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw
Warsaw, officially the Capital City of Warsaw, is the capital and List of cities and towns in Poland, largest city of Poland. The metropolis stands on the Vistula, River Vistula in east-central Poland. Its population is officially estimated at 1.86 million residents within a Warsaw metropolitan area, greater metropolitan area of 3.27 million residents, which makes Warsaw the List of cities in the European Union by population within city limits, 6th most-populous city in the European Union. The city area measures and comprises List of districts and neighbourhoods of Warsaw, 18 districts, while the metropolitan area covers . Warsaw is classified as an Globalization and World Cities Research Network#Alpha 2, alpha global city, a major political, economic and cultural hub, and the country's seat of government. It is also the capital of the Masovian Voivodeship. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th cent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lviv
Lviv ( or ; ; ; see #Names and symbols, below for other names) is the largest city in western Ukraine, as well as the List of cities in Ukraine, fifth-largest city in Ukraine, with a population of It serves as the administrative centre of Lviv Oblast and Lviv Raion, and is one of the main Ukrainian culture, cultural centres of Ukraine. Lviv also hosts the administration of Lviv urban hromada. It was named after Leo I of Galicia, the eldest son of Daniel of Galicia, Daniel, King of Ruthenia. Lviv (then Lwów) emerged as the centre of the historical regions of Red Ruthenia and Galicia (Eastern Europe), Galicia in the 14th century, superseding Halych, Chełm, Belz, and Przemyśl. It was the capital of the Kingdom of Galicia–Volhynia from 1272 to 1349, when it went to King Casimir III the Great of Kingdom of Poland, Poland in a Galicia–Volhynia Wars, war of succession. In 1356, Casimir the Great granted it town rights. From 1434, it was the regional capital of the Ruthenian ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
