1.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
2.
Baseball
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Baseball is a bat-and-ball game played between two teams of nine players each, who take turns batting and fielding. A run is scored when a player advances around the bases, Players on the batting team take turns hitting against the pitcher of the fielding team, which tries to prevent runs by getting hitters out in any of several ways. A player on the team who reaches a base safely can later attempt to advance to subsequent bases during teammates turns batting. The teams switch between batting and fielding whenever the team records three outs. One turn batting for both teams, beginning with the team, constitutes an inning. A game is composed of nine innings, and the team with the number of runs at the end of the game wins. Baseball has no clock, although almost all games end in the ninth inning. Baseball evolved from older bat-and-ball games already being played in England by the mid-18th century and this game was brought by immigrants to North America, where the modern version developed. By the late 19th century, baseball was widely recognized as the sport of the United States. Baseball is now popular in North America and parts of Central and South America, the Caribbean, in the United States and Canada, professional Major League Baseball teams are divided into the National League and American League, each with three divisions, East, West, and Central. The major league champion is determined by playoffs that culminate in the World Series, the top level of play is similarly split in Japan between the Central and Pacific Leagues and in Cuba between the West League and East League. The evolution of baseball from older bat-and-ball games is difficult to trace with precision, a French manuscript from 1344 contains an illustration of clerics playing a game, possibly la soule, with similarities to baseball. Other old French games such as thèque, la balle au bâton, consensus once held that todays baseball is a North American development from the older game rounders, popular in Great Britain and Ireland. Baseball Before We Knew It, A Search for the Roots of the Game, by David Block, suggests that the game originated in England, recently uncovered historical evidence supports this position. Block argues that rounders and early baseball were actually regional variants of other. It has long believed that cricket also descended from such games. The earliest known reference to baseball is in a 1744 British publication, A Little Pretty Pocket-Book, David Block discovered that the first recorded game of Bass-Ball took place in 1749 in Surrey, and featured the Prince of Wales as a player. William Bray, an English lawyer, recorded a game of baseball on Easter Monday 1755 in Guildford and this early form of the game was apparently brought to Canada by English immigrants
3.
Cricket
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Cricket is a bat-and-ball game played between two teams of eleven players on a cricket field, at the centre of which is a rectangular 22-yard-long pitch with a wicket at each end. One team bats, attempting to score as many runs as possible, each phase of play is called an innings. After either ten batsmen have been dismissed or a number of overs have been completed, the innings ends. The winning team is the one that scores the most runs, including any extras gained, at the start of each game, two batsmen and eleven fielders enter the field of play. The striker takes guard on a crease drawn on the four feet in front of the wicket. His role is to prevent the ball hitting the stumps by use of his bat. The other batsman, known as the non-striker, waits at the end of the pitch near the bowler. A dismissed batsman must leave the field, and a teammate replaces him, the bowlers objectives are to prevent the scoring of runs and to dismiss the batsman. An over is a set of six deliveries bowled by the same bowler, the next over is bowled from the other end of the pitch by a different bowler. If a fielder retrieves the ball enough to put down the wicket with a batsman not having reached the crease at that end of the pitch. Adjudication is performed on the field by two umpires, the laws of cricket are maintained by the International Cricket Council and the Marylebone Cricket Club. Traditionally cricketers play in all-white kit, but in limited overs cricket they wear club or team colours. In addition to the kit, some players wear protective gear to prevent injury caused by the ball. Although crickets origins are uncertain, it is first recorded in south-east England in the 16th century and it spread globally with the expansion of the British Empire, leading to the first international matches in the mid-19th century. ICC, the governing body, has over 100 members. The sport is followed primarily in Australasia, Britain, the Indian subcontinent, southern Africa, womens cricket, which is organised and played separately, has also achieved international standard. A number of words have been suggested as sources for the term cricket, in the earliest definite reference to the sport in 1598 it is called creckett. One possible source for the name is the Old English cricc or cryce meaning a crutch or staff, in Samuel Johnsons Dictionary, he derived cricket from cryce, Saxon, a stick
4.
Complex analysis
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Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. As a differentiable function of a variable is equal to the sum of its Taylor series. Complex analysis is one of the branches in mathematics, with roots in the 19th century. Important mathematicians associated with complex analysis include Euler, Gauss, Riemann, Cauchy, Weierstrass, Complex analysis, in particular the theory of conformal mappings, has many physical applications and is also used throughout analytic number theory. In modern times, it has very popular through a new boost from complex dynamics. Another important application of analysis is in string theory which studies conformal invariants in quantum field theory. A complex function is one in which the independent variable and the dependent variable are complex numbers. More precisely, a function is a function whose domain. In other words, the components of the f, u = u and v = v can be interpreted as real-valued functions of the two real variables, x and y. The basic concepts of complex analysis are often introduced by extending the elementary real functions into the complex domain, holomorphic functions are complex functions, defined on an open subset of the complex plane, that are differentiable. In the context of analysis, the derivative of f at z 0 is defined to be f ′ = lim z → z 0 f − f z − z 0, z ∈ C. Although superficially similar in form to the derivative of a real function, in particular, for this limit to exist, the value of the difference quotient must approach the same complex number, regardless of the manner in which we approach z 0 in the complex plane. Consequently, complex differentiability has much stronger consequences than usual differentiability, for instance, holomorphic functions are infinitely differentiable, whereas most real differentiable functions are not. For this reason, holomorphic functions are referred to as analytic functions. Such functions that are holomorphic everywhere except a set of isolated points are known as meromorphic functions. On the other hand, the functions z ↦ ℜ, z ↦ | z |, an important property that characterizes holomorphic functions is the relationship between the partial derivatives of their real and imaginary components, known as the Cauchy-Riemann conditions. If f, C → C, defined by f = f = u + i v, here, the differential operator ∂ / ∂ z ¯ is defined as. In terms of the real and imaginary parts of the function, u and v, this is equivalent to the pair of equations u x = v y and u y = − v x, where the subscripts indicate partial differentiation
5.
Permutation
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These differ from combinations, which are selections of some members of a set where order is disregarded. For example, written as tuples, there are six permutations of the set, namely and these are all the possible orderings of this three element set. As another example, an anagram of a word, all of whose letters are different, is a permutation of its letters, in this example, the letters are already ordered in the original word and the anagram is a reordering of the letters. The study of permutations of finite sets is a topic in the field of combinatorics, Permutations occur, in more or less prominent ways, in almost every area of mathematics. For similar reasons permutations arise in the study of sorting algorithms in computer science, the number of permutations of n distinct objects is n factorial, usually written as n. which means the product of all positive integers less than or equal to n. In algebra and particularly in group theory, a permutation of a set S is defined as a bijection from S to itself and that is, it is a function from S to S for which every element occurs exactly once as an image value. This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f, the collection of such permutations form a group called the symmetric group of S. The key to this structure is the fact that the composition of two permutations results in another rearrangement. Permutations may act on structured objects by rearranging their components, or by certain replacements of symbols, in elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set, fabian Stedman in 1677 described factorials when explaining the number of permutations of bells in change ringing. Starting from two bells, first, two must be admitted to be varied in two ways which he illustrates by showing 12 and 21 and he then explains that with three bells there are three times two figures to be produced out of three which again is illustrated. His explanation involves cast away 3, and 1.2 will remain, cast away 2, and 1.3 will remain, cast away 1, and 2.3 will remain. He then moves on to four bells and repeats the casting away argument showing that there will be four different sets of three, effectively this is an recursive process. He continues with five bells using the casting method and tabulates the resulting 120 combinations. At this point he gives up and remarks, Now the nature of these methods is such, in modern mathematics there are many similar situations in which understanding a problem requires studying certain permutations related to it. There are two equivalent common ways of regarding permutations, sometimes called the active and passive forms, or in older terminology substitutions and permutations, which form is preferable depends on the type of questions being asked in a given discipline. The active way to regard permutations of a set S is to them as the bijections from S to itself. Thus, the permutations are thought of as functions which can be composed with each other, forming groups of permutations
6.
Exponential function
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In mathematics, an exponential function is a function of the form in which the input variable x occurs as an exponent. A function of the form f = b x + c, as functions of a real variable, exponential functions are uniquely characterized by the fact that the growth rate of such a function is directly proportional to the value of the function. The constant of proportionality of this relationship is the logarithm of the base b. The argument of the function can be any real or complex number or even an entirely different kind of mathematical object. Its ubiquitous occurrence in pure and applied mathematics has led mathematician W. Rudin to opine that the function is the most important function in mathematics. In applied settings, exponential functions model a relationship in which a constant change in the independent variable gives the same change in the dependent variable. The graph of y = e x is upward-sloping, and increases faster as x increases, the graph always lies above the x -axis but can get arbitrarily close to it for negative x, thus, the x -axis is a horizontal asymptote. The slope of the tangent to the graph at each point is equal to its y -coordinate at that point, as implied by its derivative function. Its inverse function is the logarithm, denoted log, ln, or log e, because of this. The exponential function exp, C → C can be characterized in a variety of equivalent ways, the constant e is then defined as e = exp = ∑ k =0 ∞. The exponential function arises whenever a quantity grows or decays at a proportional to its current value. One such situation is continuously compounded interest, and in fact it was this observation that led Jacob Bernoulli in 1683 to the number lim n → ∞ n now known as e, later, in 1697, Johann Bernoulli studied the calculus of the exponential function. If instead interest is compounded daily, this becomes 365, letting the number of time intervals per year grow without bound leads to the limit definition of the exponential function, exp = lim n → ∞ n first given by Euler. This is one of a number of characterizations of the exponential function, from any of these definitions it can be shown that the exponential function obeys the basic exponentiation identity, exp = exp ⋅ exp which is why it can be written as ex. The derivative of the function is the exponential function itself. More generally, a function with a rate of change proportional to the function itself is expressible in terms of the exponential function and this function property leads to exponential growth and exponential decay. The exponential function extends to a function on the complex plane. Eulers formula relates its values at purely imaginary arguments to trigonometric functions, the exponential function also has analogues for which the argument is a matrix, or even an element of a Banach algebra or a Lie algebra
7.
Real-valued function
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In mathematics, a real-valued function or real function is a function whose values are real numbers. In other words, it is a function that assigns a number to each member of its domain. Many important function spaces are defined to consist of real functions, let X be an arbitrary set. Let F denote the set of all functions from X to real numbers R. F is an ordered ring. The σ-algebra of Borel sets is an important structure on real numbers, if X has its σ-algebra and a function f is such that the preimage f −1 of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a space and an algebra as explained above. Moreover, a set of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets and this is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space Ω are real-valued random variables. Real numbers form a space and a complete metric space. Continuous real-valued functions are important in theories of topological spaces and of metric spaces, the extreme value theorem states that for any real continuous function on a compact space its global maximum and minimum exist. The concept of space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a topological space. Continuous functions also form a space and an algebra as explained above. Real numbers are used as the codomain to define smooth functions, a domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold. Spaces of smooth functions also are vector spaces and algebras as explained above, a measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. Lp spaces on sets with a measure are defined from aforementioned real-valued measurable functions, though, real-valued Lp spaces still have some of the structure explicated above. For example, pointwise product of two L2 functions belongs to L1, Real analysis Partial differential equations, a major user of real-valued functions Norm Scalar Weisstein, Eric W. Real Function
8.
Batting order (baseball)
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In baseball, the batting order or batting lineup is the sequence in which the members of the offense take their turns in batting against the pitcher. The batting order is the component of a teams offensive strategy. The home plate umpire keeps one copy of the card of each team. Once the home plate umpire gives the lineup cards to the managers, the batting lineup is final. If a team out of order, it is a violation of baseballs rules. Dictionary. com, however, defines bat around as to have every player in the lineup take a turn at bat during a single inning and it is not an official statistic. Opinions differ as to whether nine batters must get an at-bat, in modern American baseball, some batting positions have nicknames, leadoff for first, cleanup for fourth, and last for ninth. Others are known by the numbers or the term #-hole. At the start of each inning, the batting order resumes where it left off in the previous inning, early forms of baseball or rounders from the mid 19th-Century did not require a fixed batting order, any player who was not on base could be called upon to bat. The concept of a set batting order is said to have been invented by Alexander Cartwright, who also instituted rules such as the ball and tagging the runner. In the early days of baseball, the rules did not require that the order be announced before game time. This permitted strategic decisions regarding batting order to occur while the game was in progress, for example, Cap Anson was known to wait to see if the first two men got on base in the first inning. If they did not, he would wait and hit in the next inning, however, in the 1880s, organized baseball began mandating that the batting order be disclosed before the first pitch. For example, Rule 36 in The Playing Rules of Professional Base Ball Clubs of 1896 stated the following, after the first inning the first striker in each inning shall be the batsman whose name follows that of the last man who completed his turn. In cricket, the order is generally fixed so that players are sure of their role within the team. A batsman can be promoted to a spot in the batting order according to the teams wishes. The idea of a batting order, in which the on-deck batter at the time the final out is made in one inning becomes the lead-off batter in the next inning, is unique to baseball. In the shorter form of cricket, there is only one innings per side, in a typical innings of this latter form, all eleven players on the team will have a chance to bat, and the innings finishes when 10 players are out
9.
Inverse function
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I. e. f = y if and only if g = x. As a simple example, consider the function of a real variable given by f = 5x −7. Thinking of this as a procedure, to reverse this and get x back from some output value, say y. In this case means that we should add 7 to y. In functional notation this inverse function would be given by, g = y +75, with y = 5x −7 we have that f = y and g = x. Not all functions have inverse functions, in order for a function f, X → Y to have an inverse, it must have the property that for every y in Y there must be one, and only one x in X so that f = y. This property ensures that a function g, Y → X will exist having the necessary relationship with f, let f be a function whose domain is the set X, and whose image is the set Y. Then f is invertible if there exists a g with domain Y and image X, with the property. If f is invertible, the g is unique, which means that there is exactly one function g satisfying this property. That function g is called the inverse of f, and is usually denoted as f −1. Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, not all functions have an inverse. For a function to have an inverse, each element y ∈ Y must correspond to no more than one x ∈ X, a function f with this property is called one-to-one or an injection. If f −1 is to be a function on Y, then each element y ∈ Y must correspond to some x ∈ X. Functions with this property are called surjections. This property is satisfied by definition if Y is the image of f, to be invertible a function must be both an injection and a surjection. If a function f is invertible, then both it and its inverse function f−1 are bijections, there is another convention used in the definition of functions. This can be referred to as the set-theoretic or graph definition using ordered pairs in which a codomain is never referred to, under this convention all functions are surjections, and so, being a bijection simply means being an injection. Authors using this convention may use the phrasing that a function is invertible if, the two conventions need not cause confusion as long as it is remembered that in this alternate convention the codomain of a function is always taken to be the range of the function. With this type of function it is impossible to deduce an input from its output, such a function is called non-injective or, in some applications, information-losing
10.
Unicode
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Unicode is a computing industry standard for the consistent encoding, representation, and handling of text expressed in most of the worlds writing systems. As of June 2016, the most recent version is Unicode 9.0, the standard is maintained by the Unicode Consortium. Unicodes success at unifying character sets has led to its widespread, the standard has been implemented in many recent technologies, including modern operating systems, XML, Java, and the. NET Framework. Unicode can be implemented by different character encodings, the most commonly used encodings are UTF-8, UTF-16 and the now-obsolete UCS-2. UTF-8 uses one byte for any ASCII character, all of which have the same values in both UTF-8 and ASCII encoding, and up to four bytes for other characters. UCS-2 uses a 16-bit code unit for each character but cannot encode every character in the current Unicode standard, UTF-16 extends UCS-2, using one 16-bit unit for the characters that were representable in UCS-2 and two 16-bit units to handle each of the additional characters. Many traditional character encodings share a common problem in that they allow bilingual computer processing, Unicode, in intent, encodes the underlying characters—graphemes and grapheme-like units—rather than the variant glyphs for such characters. In the case of Chinese characters, this leads to controversies over distinguishing the underlying character from its variant glyphs. In text processing, Unicode takes the role of providing a unique code point—a number, in other words, Unicode represents a character in an abstract way and leaves the visual rendering to other software, such as a web browser or word processor. This simple aim becomes complicated, however, because of concessions made by Unicodes designers in the hope of encouraging a more rapid adoption of Unicode, the first 256 code points were made identical to the content of ISO-8859-1 so as to make it trivial to convert existing western text. For other examples, see duplicate characters in Unicode and he explained that he name Unicode is intended to suggest a unique, unified, universal encoding. In this document, entitled Unicode 88, Becker outlined a 16-bit character model, Unicode could be roughly described as wide-body ASCII that has been stretched to 16 bits to encompass the characters of all the worlds living languages. In a properly engineered design,16 bits per character are more than sufficient for this purpose, Unicode aims in the first instance at the characters published in modern text, whose number is undoubtedly far below 214 =16,384. By the end of 1990, most of the work on mapping existing character encoding standards had been completed, the Unicode Consortium was incorporated in California on January 3,1991, and in October 1991, the first volume of the Unicode standard was published. The second volume, covering Han ideographs, was published in June 1992, in 1996, a surrogate character mechanism was implemented in Unicode 2.0, so that Unicode was no longer restricted to 16 bits. The Microsoft TrueType specification version 1.0 from 1992 used the name Apple Unicode instead of Unicode for the Platform ID in the naming table, Unicode defines a codespace of 1,114,112 code points in the range 0hex to 10FFFFhex. Normally a Unicode code point is referred to by writing U+ followed by its hexadecimal number, for code points in the Basic Multilingual Plane, four digits are used, for code points outside the BMP, five or six digits are used, as required. Code points in Planes 1 through 16 are accessed as surrogate pairs in UTF-16, within each plane, characters are allocated within named blocks of related characters
11.
Function (mathematics)
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In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that each real number x to its square x2. The output of a function f corresponding to a x is denoted by f. In this example, if the input is −3, then the output is 9, likewise, if the input is 3, then the output is also 9, and we may write f =9. The input variable are sometimes referred to as the argument of the function, Functions of various kinds are the central objects of investigation in most fields of modern mathematics. There are many ways to describe or represent a function, some functions may be defined by a formula or algorithm that tells how to compute the output for a given input. Others are given by a picture, called the graph of the function, in science, functions are sometimes defined by a table that gives the outputs for selected inputs. A function could be described implicitly, for example as the inverse to another function or as a solution of a differential equation, sometimes the codomain is called the functions range, but more commonly the word range is used to mean, instead, specifically the set of outputs. For example, we could define a function using the rule f = x2 by saying that the domain and codomain are the numbers. The image of this function is the set of real numbers. In analogy with arithmetic, it is possible to define addition, subtraction, multiplication, another important operation defined on functions is function composition, where the output from one function becomes the input to another function. Linking each shape to its color is a function from X to Y, each shape is linked to a color, there is no shape that lacks a color and no shape that has more than one color. This function will be referred to as the color-of-the-shape function, the input to a function is called the argument and the output is called the value. The set of all permitted inputs to a function is called the domain of the function. Thus, the domain of the function is the set of the four shapes. The concept of a function does not require that every possible output is the value of some argument, a second example of a function is the following, the domain is chosen to be the set of natural numbers, and the codomain is the set of integers. The function associates to any number n the number 4−n. For example, to 1 it associates 3 and to 10 it associates −6, a third example of a function has the set of polygons as domain and the set of natural numbers as codomain
12.
Smoothness
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In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has which are continuous. A smooth function is a function that has derivatives of all orders everywhere in its domain, differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives, consider an open set on the real line and a function f defined on that set with real values. Let k be a non-negative integer, the function f is said to be of class Ck if the derivatives f′, f′′. The function f is said to be of class C∞, or smooth, if it has derivatives of all orders. The function f is said to be of class Cω, or analytic, if f is smooth, Cω is thus strictly contained in C∞. Bump functions are examples of functions in C∞ but not in Cω, to put it differently, the class C0 consists of all continuous functions. The class C1 consists of all differentiable functions whose derivative is continuous, thus, a C1 function is exactly a function whose derivative exists and is of class C0. In particular, Ck is contained in Ck−1 for every k, C∞, the class of infinitely differentiable functions, is the intersection of the sets Ck as k varies over the non-negative integers. The function f = { x if x ≥0,0 if x <0 is continuous, because cos oscillates as x →0, f ’ is not continuous at zero. Therefore, this function is differentiable but not of class C1, the functions f = | x | k +1 where k is even, are continuous and k times differentiable at all x. But at x =0 they are not times differentiable, so they are of class Ck, the exponential function is analytic, so, of class Cω. The trigonometric functions are also analytic wherever they are defined, the function f is an example of a smooth function with compact support. Let n and m be some positive integers, if f is a function from an open subset of Rn with values in Rm, then f has component functions f1. Each of these may or may not have partial derivatives, the classes C∞ and Cω are defined as before. These criteria of differentiability can be applied to the functions of a differential structure. The resulting space is called a Ck manifold, if one wishes to start with a coordinate-independent definition of the class Ck, one may start by considering maps between Banach spaces. A map from one Banach space to another is differentiable at a point if there is a map which approximates it at that point