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113-476: If all numbers in a list are the same number, then their average is also equal to this number. This property is shared by each of the many types of average. Another universal property is monotonicity : if two lists of numbers A and B have the same length, and each entry of list A is at least as large as the corresponding entry on list B , then the average of list A is at least that of list B . Also, all averages satisfy linear homogeneity : if all numbers of

226-645: A δ > 0 {\displaystyle \delta >0} such that for all x ∈ D {\displaystyle x\in D} : | x − x 0 | < δ      implies      | f ( x ) − f ( x 0 ) | < ε . {\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .} More intuitively, we can say that if we want to get all

339-413: A monotonic function (or monotone function ) is a function between ordered sets that preserves or reverses the given order . This concept first arose in calculus , and was later generalized to the more abstract setting of order theory . In calculus , a function f {\displaystyle f} defined on a subset of the real numbers with real values is called monotonic if it

452-417: A time series , such as daily stock market prices or yearly temperatures, people often want to create a smoother series. This helps to show underlying trends or perhaps periodic behavior. An easy way to do this is the moving average : one chooses a number n and creates a new series by taking the arithmetic mean of the first n values, then moving forward one place by dropping the oldest value and introducing

565-679: A (possibly non-linear) operator T : X → X ∗ {\displaystyle T:X\rightarrow X^{*}} is said to be a monotone operator if ( T u − T v , u − v ) ≥ 0 ∀ u , v ∈ X . {\displaystyle (Tu-Tv,u-v)\geq 0\quad \forall u,v\in X.} Kachurovskii's theorem shows that convex functions on Banach spaces have monotonic operators as their derivatives. A subset G {\displaystyle G} of X × X ∗ {\displaystyle X\times X^{*}}

678-461: A , b , c , since it can be written for instance as (( a and b ) or ( a and c ) or ( b and c )). The number of such functions on n variables is known as the Dedekind number of n . SAT solving , generally an NP-hard task, can be achieved efficiently when all involved functions and predicates are monotonic and Boolean. Continuous function In mathematics , a continuous function

791-437: A Romance origin. Due to the aforementioned colloquial nature of the term "average", the term can be used to obfuscate the true meaning of data and suggest varying answers to questions based on the averaging method (most frequently arithmetic mean, median, or mode) used. In his article "Framed for Lying: Statistics as In/Artistic Proof", University of Pittsburgh faculty member Daniel Libertz comments that statistical information

904-400: A common method to use for reducing errors of measurement in various areas. At the time, astronomers wanted to know a real value from noisy measurement, such as the position of a planet or the diameter of the moon. Using the mean of several measured values, scientists assumed that the errors add up to a relatively small number when compared to the total of all measured values. The method of taking

1017-527: A continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous. One can instead require that for any sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} of points in

1130-399: A function f is continuous at a point x 0 {\displaystyle x_{0}} if and only if its oscillation at that point is zero; in symbols, ω f ( x 0 ) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much

1243-443: A function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point f ( c ) {\displaystyle f(c)} as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood N 1 ( f ( c ) ) {\displaystyle N_{1}(f(c))} there

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1356-488: A generalization of real numbers. The above definition of monotonicity is relevant in these cases as well. However, the terms "increasing" and "decreasing" are avoided, since their conventional pictorial representation does not apply to orders that are not total . Furthermore, the strict relations < {\displaystyle <} and > {\displaystyle >} are of little use in many non-total orders and hence no additional terminology

1469-404: A list are multiplied by the same positive number, then its average changes by the same factor. In some types of average, the items in the list are assigned different weights before the average is determined. These include the weighted arithmetic mean , the weighted geometric mean and the weighted median . Also, for some types of moving average , the weight of an item depends on its position in

1582-422: A list of arguments that is continuous , strictly increasing in each argument, and symmetric (invariant under permutation of the arguments). The average y is then the value that, when replacing each member of the list, results in the same function value: g ( y , y , ..., y ) = g ( x 1 , x 2 , ..., x n ) . This most general definition still captures the important property of all averages that

1695-460: A merchant sea voyage"; and the same meaning for avaria is in Marseille in 1210, Barcelona in 1258 and Florence in the late 13th. 15th-century French avarie had the same meaning, and it begot English "averay" (1491) and English "average" (1502) with the same meaning. Today, Italian avaria , Catalan avaria and French avarie still have the primary meaning of "damage". The huge transformation of

1808-486: A monotonic transform (see also monotone preferences ). In this context, the term "monotonic transformation" refers to a positive monotonic transformation and is intended to distinguish it from a "negative monotonic transformation," which reverses the order of the numbers. The following properties are true for a monotonic function f : R → R {\displaystyle f\colon \mathbb {R} \to \mathbb {R} } : These properties are

1921-563: A neighbourhood N ( x 0 ) {\textstyle N(x_{0})} that | f ( x ) − f ( x 0 ) | ≤ C ( | x − x 0 | )  for all  x ∈ D ∩ N ( x 0 ) {\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})} A function

2034-404: A new value at the other end of the list, and so on. This is the simplest form of moving average. More complicated forms involve using a weighted average . The weighting can be used to enhance or suppress various periodic behavior and there is very extensive analysis of what weightings to use in the literature on filtering . In digital signal processing the term "moving average" is used even when

2147-451: A one-to-one mapping from their range to their domain. However, functions that are only weakly monotone are not invertible because they are constant on some interval (and therefore are not one-to-one). A function may be strictly monotonic over a limited a range of values and thus have an inverse on that range even though it is not strictly monotonic everywhere. For example, if y = g ( x ) {\displaystyle y=g(x)}

2260-489: A rapid proof of one direction of the Lebesgue integrability condition . The oscillation is equivalent to the ε − δ {\displaystyle \varepsilon -\delta } definition by a simple re-arrangement and by using a limit ( lim sup , lim inf ) to define oscillation: if (at a given point) for a given ε 0 {\displaystyle \varepsilon _{0}} there

2373-457: A similar vein, Dirichlet's function , the indicator function for the set of rational numbers, D ( x ) = { 0  if  x  is irrational  ( ∈ R ∖ Q ) 1  if  x  is rational  ( ∈ Q ) {\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{

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2486-422: A source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible. The term monotonic transformation (or monotone transformation ) may also cause confusion because it refers to a transformation by a strictly increasing function. This is the case in economics with respect to the ordinal properties of a utility function being preserved across

2599-1078: Is C {\displaystyle C} -continuous for some C ∈ C . {\displaystyle C\in {\mathcal {C}}.} For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions C L i p s c h i t z = { C : C ( δ ) = K | δ | ,   K > 0 } {\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}} respectively C Hölder − α = { C : C ( δ ) = K | δ | α ,   K > 0 } . {\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}.} Continuity can also be defined in terms of oscillation :

2712-532: Is not continuous . 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

2825-411: Is continuous at the real number c , if the limit of f ( x ) , {\displaystyle f(x),} as x tends to c , is equal to f ( c ) . {\displaystyle f(c).} There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain . A function is continuous on an open interval if

2938-535: Is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and x ↦ sin ⁡ ( 1 x ) {\textstyle x\mapsto \sin({\frac {1}{x}})} are discontinuous at 0 , and remain discontinuous whichever value

3051-439: Is +60%, then the average percentage return or CAGR, R , can be obtained by solving the equation: (1 − 10%) × (1 + 60%) = (1 − 0.1) × (1 + 0.6) = (1 + R ) × (1 + R ) . The value of R that makes this equation true is 0.2, or 20%. This means that the total return over the 2-year period is the same as if there had been 20% growth each year. The order of the years makes no difference – the average percentage returns of +60% and −10%

3164-456: Is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below. Continuity of real functions is usually defined in terms of limits . A function f with variable x

3277-406: 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

3390-468: Is a desired δ , {\displaystyle \delta ,} the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space . Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse , page 34). Non-standard analysis

3503-468: Is a monotonically increasing function. A function is unimodal if it is monotonically increasing up to some point (the mode ) and then monotonically decreasing. When f {\displaystyle f} is a strictly monotonic function, then f {\displaystyle f} is injective on its domain, and if T {\displaystyle T} is the range of f {\displaystyle f} , then there

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3616-561: Is a neighborhood N 2 ( c ) {\displaystyle N_{2}(c)} in its domain such that f ( x ) ∈ N 1 ( f ( c ) ) {\displaystyle f(x)\in N_{1}(f(c))} whenever x ∈ N 2 ( c ) . {\displaystyle x\in N_{2}(c).} As neighborhoods are defined in any topological space , this definition of

3729-426: Is a rational number 0  if  x  is irrational . {\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}} is continuous at all irrational numbers and discontinuous at all rational numbers. In

3842-475: Is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers . In nonstandard analysis, continuity can be defined as follows. (see microcontinuity ). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy 's definition of continuity. Checking

3955-474: Is also admissible , monotonicity is a stricter requirement than admissibility. Some heuristic algorithms such as A* can be proven optimal provided that the heuristic they use is monotonic. In Boolean algebra , a monotonic function is one such that for all a i and b i in {0,1} , if a 1 ≤ b 1 , a 2 ≤ b 2 , ..., a n ≤ b n (i.e. the Cartesian product {0, 1}

4068-459: Is also continuous on D ∖ { x : g ( x ) = 0 } {\displaystyle D\setminus \{x:g(x)=0\}} . For example, the function (pictured) y ( x ) = 2 x − 1 x + 2 {\displaystyle y(x)={\frac {2x-1}{x+2}}} is defined for all real numbers x ≠ − 2 {\displaystyle x\neq -2} and

4181-439: Is also monotone. The dual notion is often called antitone , anti-monotone , or order-reversing . Hence, an antitone function f satisfies the property x ≤ y ⟹ f ( y ) ≤ f ( x ) , {\displaystyle x\leq y\implies f(y)\leq f(x),} for all x and y in its domain. A constant function is both monotone and antitone; conversely, if f

4294-472: Is an inverse function on T {\displaystyle T} for f {\displaystyle f} . In contrast, each constant function is monotonic, but not injective, and hence cannot have an inverse. The graphic shows six monotonic functions. Their simplest forms are shown in the plot area and the expressions used to create them are shown on the y -axis. A map f : X → Y {\displaystyle f:X\to Y}

4407-604: Is both monotone and antitone, and if the domain of f is a lattice , then f must be constant. Monotone functions are central in order theory. They appear in most articles on the subject and examples from special applications are found in these places. Some notable special monotone functions are order embeddings (functions for which x ≤ y {\displaystyle x\leq y} if and only if f ( x ) ≤ f ( y ) ) {\displaystyle f(x)\leq f(y))} and order isomorphisms ( surjective order embeddings). In

4520-826: Is called strictly monotone . Functions that are strictly monotone are one-to-one (because for x {\displaystyle x} not equal to y {\displaystyle y} , either x < y {\displaystyle x<y} or x > y {\displaystyle x>y} and so, by monotonicity, either f ( x ) < f ( y ) {\displaystyle f\!\left(x\right)<f\!\left(y\right)} or f ( x ) > f ( y ) {\displaystyle f\!\left(x\right)>f\!\left(y\right)} , thus f ( x ) ≠ f ( y ) {\displaystyle f\!\left(x\right)\neq f\!\left(y\right)} .) To avoid ambiguity,

4633-411: Is chosen for defining them at 0 . A point where a function is discontinuous is called a discontinuity . Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above. Let f : D → R {\displaystyle f:D\to \mathbb {R} } be a function defined on a subset D {\displaystyle D} of

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4746-666: Is continuous at every such point. Thus, it is a continuous function. The question of continuity at x = − 2 {\displaystyle x=-2} does not arise since x = − 2 {\displaystyle x=-2} is not in the domain of y . {\displaystyle y.} There is no continuous function F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } that agrees with y ( x ) {\displaystyle y(x)} for all x ≠ − 2. {\displaystyle x\neq -2.} Since

4859-460: Is continuous everywhere apart from x = 0 {\displaystyle x=0} . Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological , for example, Thomae's function , f ( x ) = { 1  if  x = 0 1 q  if  x = p q (in lowest terms)

4972-462: Is continuous in x 0 {\displaystyle x_{0}} if it is C -continuous for some control function C . This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions C {\displaystyle {\mathcal {C}}} a function is C {\displaystyle {\mathcal {C}}} -continuous if it

5085-680: Is continuous in D ∖ { x : f ( x ) = 0 } . {\displaystyle D\setminus \{x:f(x)=0\}.} This implies that, excluding the roots of g , {\displaystyle g,} the quotient of continuous functions q = f / g {\displaystyle q=f/g} (defined by q ( x ) = f ( x ) / g ( x ) {\displaystyle q(x)=f(x)/g(x)} for all x ∈ D {\displaystyle x\in D} , such that g ( x ) ≠ 0 {\displaystyle g(x)\neq 0} )

5198-488: Is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}}

5311-442: Is continuous on its whole domain, which is the closed interval [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).} Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points . Examples include the reciprocal function x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and

5424-874: Is continuous. This construction allows stating, for example, that e sin ⁡ ( ln ⁡ x ) {\displaystyle e^{\sin(\ln x)}} is continuous for all x > 0. {\displaystyle x>0.} An example of a discontinuous function is the Heaviside step function H {\displaystyle H} , defined by H ( x ) = { 1  if  x ≥ 0 0  if  x < 0 {\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}} Pick for instance ε = 1 / 2 {\displaystyle \varepsilon =1/2} . Then there

5537-500: Is discontinuous at x = 0 {\displaystyle x=0} but continuous everywhere else. Yet another example: the function f ( x ) = { sin ⁡ ( x − 2 )  if  x ≠ 0 0  if  x = 0 {\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}

5650-470: Is earlier (from at least the 11th century), unrelated use of the word. It appears to be an old legal term for a tenant's day labour obligation to a sheriff, probably anglicised from "avera" found in the English Domesday Book (1085). The Oxford English Dictionary, however, says that derivations from German hafen haven, and Arabic ʿawâr loss, damage, have been "quite disposed of" and the word has

5763-686: Is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed monotonically increasing (also increasing or non-decreasing ) if for all x {\displaystyle x} and y {\displaystyle y} such that x ≤ y {\displaystyle x\leq y} one has f ( x ) ≤ f ( y ) {\displaystyle f\!\left(x\right)\leq f\!\left(y\right)} , so f {\displaystyle f} preserves

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5876-686: Is frequently dismissed from rhetorical arguments for this reason. However, due to their persuasive power, averages and other statistical values should not be discarded completely, but instead used and interpreted with caution. Libertz invites us to engage critically not only with statistical information such as averages, but also with the language used to describe the data and its uses, saying: "If statistics rely on interpretation, rhetors should invite their audience to interpret rather than insist on an interpretation." In many cases, data and specific calculations are provided to help facilitate this audience-based interpretation. Monotonicity In mathematics ,

5989-491: Is introduced for them. Letting ≤ {\displaystyle \leq } denote the partial order relation of any partially ordered set, a monotone function, also called isotone , or order-preserving , satisfies the property x ≤ y ⟹ f ( x ) ≤ f ( y ) {\displaystyle x\leq y\implies f(x)\leq f(y)} for all x and y in its domain. The composite of two monotone mappings

6102-443: Is left, it is the median; if two values, the median is the arithmetic mean of these two. This method takes the list 1, 7, 3, 13 and orders it to read 1, 3, 7, 13. Then the 1 and 13 are removed to obtain the list 3, 7. Since there are two elements in this remaining list, the median is their arithmetic mean, (3 + 7)/2 = 5. The mid-range is the arithmetic mean of the highest and lowest values of a set. Even though perhaps not an average,

6215-487: Is neither non-decreasing nor non-increasing. A function f {\displaystyle f} is said to be absolutely monotonic over an interval ( a , b ) {\displaystyle \left(a,b\right)} if the derivatives of all orders of f {\displaystyle f} are nonnegative or all nonpositive at all points on the interval. All strictly monotonic functions are invertible because they are guaranteed to have

6328-415: Is no δ {\displaystyle \delta } that satisfies the ε − δ {\displaystyle \varepsilon -\delta } definition, then the oscillation is at least ε 0 , {\displaystyle \varepsilon _{0},} and conversely if for every ε {\displaystyle \varepsilon } there

6441-480: Is no δ {\displaystyle \delta } -neighborhood around x = 0 {\displaystyle x=0} , i.e. no open interval ( − δ , δ ) {\displaystyle (-\delta ,\;\delta )} with δ > 0 , {\displaystyle \delta >0,} that will force all the H ( x ) {\displaystyle H(x)} values to be within

6554-477: Is no agreed definition of mode. Some authors say they are all modes and some say there is no mode. The median is the middle number of the group when they are ranked in order. (If there are an even number of numbers, the mean of the middle two is taken.) Thus to find the median, order the list according to its elements' magnitude and then repeatedly remove the pair consisting of the highest and lowest values until either one or two values are left. If exactly one value

6667-524: Is ordered coordinatewise ), then f( a 1 , ..., a n ) ≤ f( b 1 , ..., b n ) . In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that an n -ary Boolean function is monotonic when its representation as an n -cube labelled with truth values has no upward edge from true to false . (This labelled Hasse diagram

6780-403: Is said to be maximal monotone if it is maximal among all monotone sets in the sense of set inclusion. The graph of a monotone operator G ( T ) {\displaystyle G(T)} is a monotone set. A monotone operator is said to be maximal monotone if its graph is a maximal monotone set . Order theory deals with arbitrary partially ordered sets and preordered sets as

6893-472: Is said to be monotone if each of its fibers is connected ; that is, for each element y ∈ Y , {\displaystyle y\in Y,} the (possibly empty) set f − 1 ( y ) {\displaystyle f^{-1}(y)} is a connected subspace of X . {\displaystyle X.} In functional analysis on a topological vector space X {\displaystyle X} ,

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7006-553: Is said to be a monotone set if for every pair [ u 1 , w 1 ] {\displaystyle [u_{1},w_{1}]} and [ u 2 , w 2 ] {\displaystyle [u_{2},w_{2}]} in G {\displaystyle G} , ( w 1 − w 2 , u 1 − u 2 ) ≥ 0. {\displaystyle (w_{1}-w_{2},u_{1}-u_{2})\geq 0.} G {\displaystyle G}

7119-403: Is strictly increasing on the range [ a , b ] {\displaystyle [a,b]} , then it has an inverse x = h ( y ) {\displaystyle x=h(y)} on the range [ g ( a ) , g ( b ) ] {\displaystyle [g(a),g(b)]} . The term monotonic is sometimes used in place of strictly monotonic , so

7232-417: Is the dual of the function's labelled Venn diagram , which is the more common representation for n ≤ 3 .) The monotonic Boolean functions are precisely those that can be defined by an expression combining the inputs (which may appear more than once) using only the operators and and or (in particular not is forbidden). For instance "at least two of a , b , c hold" is a monotonic function of

7345-503: Is the same result as that for −10% and +60%. This method can be generalized to examples in which the periods are not equal. For example, consider a period of a half of a year for which the return is −23% and a period of two and a half years for which the return is +13%. The average percentage return for the combined period is the single year return, R , that is the solution of the following equation: (1 − 0.23) × (1 + 0.13) = (1 + R ) , giving an average return R of 0.0600 or 6.00%. Given

7458-846: The ε {\displaystyle \varepsilon } -neighborhood of H ( 0 ) {\displaystyle H(0)} , i.e. within ( 1 / 2 , 3 / 2 ) {\displaystyle (1/2,\;3/2)} . Intuitively, we can think of this type of discontinuity as a sudden jump in function values. Similarly, the signum or sign function sgn ⁡ ( x ) = {   1  if  x > 0   0  if  x = 0 − 1  if  x < 0 {\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}

7571-540: The τ {\displaystyle \tau } th quantile (another summary statistic that generalizes the median) can similarly be expressed as a solution to the optimization problem which aims to minimize the total tilted absolute value loss (or quantile loss or pinball loss). The table of mathematical symbols explains the symbols used below. Other more sophisticated averages are: trimean , trimedian , and normalized mean , with their generalizations. One can create one's own average metric using

7684-427: The f ( x 0 ) {\displaystyle f(x_{0})} neighborhood is, then f {\displaystyle f} is continuous at x 0 . {\displaystyle x_{0}.} In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology , here the metric topology . Weierstrass had required that

7797-441: The f ( x ) {\displaystyle f(x)} values to stay in some small neighborhood around f ( x 0 ) , {\displaystyle f\left(x_{0}\right),} we need to choose a small enough neighborhood for the x {\displaystyle x} values around x 0 . {\displaystyle x_{0}.} If we can do that no matter how small

7910-430: The product of continuous functions , p = f ⋅ g {\displaystyle p=f\cdot g} (defined by p ( x ) = f ( x ) ⋅ g ( x ) {\displaystyle p(x)=f(x)\cdot g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} Combining

8023-409: The sum of continuous functions s = f + g {\displaystyle s=f+g} (defined by s ( x ) = f ( x ) + g ( x ) {\displaystyle s(x)=f(x)+g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} The same holds for

8136-413: The generalized f -mean : where f is any invertible function. The harmonic mean is an example of this using f ( x ) = 1/ x , and the geometric mean is another, using f ( x ) = log  x . However, this method for generating means is not general enough to capture all averages. A more general method for defining an average takes any function g ( x 1 ,  x 2 , ...,  x n ) of

8249-509: The mid-range are often used in addition to the mean as estimates of central tendency in descriptive statistics . These can all be seen as minimizing variation by some measure; see Central tendency § Solutions to variational problems . The most frequently occurring number in a list is called the mode. For example, the mode of the list (1, 2, 2, 3, 3, 3, 4) is 3. It may happen that there are two or more numbers which occur equally often and more often than any other number. In this case there

8362-417: The tangent function x ↦ tan ⁡ x . {\displaystyle x\mapsto \tan x.} When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are discontinuous. A partial function

8475-561: The above preservations of continuity and the continuity of constant functions and of the identity function I ( x ) = x {\displaystyle I(x)=x} on R {\displaystyle \mathbb {R} } , one arrives at the continuity of all polynomial functions on R {\displaystyle \mathbb {R} } , such as f ( x ) = x 3 + x 2 − 5 x + 3 {\displaystyle f(x)=x^{3}+x^{2}-5x+3} (pictured on

8588-490: The amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y = f ( x ) {\displaystyle y=f(x)} as follows: an infinitely small increment α {\displaystyle \alpha } of

8701-465: The average of a list of identical elements is that element itself. The function g ( x 1 , x 2 , ..., x n ) = x 1 + x 2 + ··· + x n provides the arithmetic mean. The function g ( x 1 , x 2 , ..., x n ) = x 1 x 2 ··· x n (where the list elements are positive numbers) provides the geometric mean. The function g ( x 1 , x 2 , ..., x n ) = ( x 1 + x 2 + ··· + x n )) (where

8814-678: The average, although there seem to be no direct record of the calculation. The root is found in Arabic as عوار ʿawār , a defect, or anything defective or damaged, including partially spoiled merchandise; and عواري ʿawārī (also عوارة ʿawāra ) = "of or relating to ʿawār , a state of partial damage". Within the Western languages the word's history begins in medieval sea-commerce on the Mediterranean. 12th and 13th century Genoa Latin avaria meant "damage, loss and non-normal expenses arising in connection with

8927-402: The context of search algorithms monotonicity (also called consistency) is a condition applied to heuristic functions . A heuristic h ( n ) {\displaystyle h(n)} is monotonic if, for every node n and every successor n' of n generated by any action a , the estimated cost of reaching the goal from n is no greater than the step cost of getting to n' plus

9040-409: The continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given f , g : D → R , {\displaystyle f,g\colon D\to \mathbb {R} ,} then

9153-416: The definition of the limit of a function, we obtain a self-contained definition: Given a function f : D → R {\displaystyle f:D\to \mathbb {R} } as above and an element x 0 {\displaystyle x_{0}} of the domain D {\displaystyle D} , f {\displaystyle f} is said to be continuous at

9266-940: The domain of f {\displaystyle f} with x 0 − δ < x < x 0 + δ , {\displaystyle x_{0}-\delta <x<x_{0}+\delta ,} the value of f ( x ) {\displaystyle f(x)} satisfies f ( x 0 ) − ε < f ( x ) < f ( x 0 ) + ε . {\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .} Alternatively written, continuity of f : D → R {\displaystyle f:D\to \mathbb {R} } at x 0 ∈ D {\displaystyle x_{0}\in D} means that for every ε > 0 , {\displaystyle \varepsilon >0,} there exists

9379-409: The domain of f , exists and is equal to f ( c ) . {\displaystyle f(c).} In mathematical notation, this is written as lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}{f(x)}=f(c).} In detail this means three conditions: first, f has to be defined at c (guaranteed by

9492-820: The domain which converges to c , the corresponding sequence ( f ( x n ) ) n ∈ N {\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }} converges to f ( c ) . {\displaystyle f(c).} In mathematical notation, ∀ ( x n ) n ∈ N ⊂ D : lim n → ∞ x n = c ⇒ lim n → ∞ f ( x n ) = f ( c ) . {\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.} Explicitly including

9605-418: The estimated cost of reaching the goal from n' , h ( n ) ≤ c ( n , a , n ′ ) + h ( n ′ ) . {\displaystyle h(n)\leq c\left(n,a,n'\right)+h\left(n'\right).} This is a form of triangle inequality , with n , n' , and the goal G n closest to n . Because every monotonic heuristic

9718-439: The function sine is continuous on all reals, the sinc function G ( x ) = sin ⁡ ( x ) / x , {\displaystyle G(x)=\sin(x)/x,} is defined and continuous for all real x ≠ 0. {\displaystyle x\neq 0.} However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining

9831-418: The function is discontinuous at a point. This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε {\displaystyle \varepsilon } (hence a G δ {\displaystyle G_{\delta }} set ) – and gives

9944-422: The function to be defined only at and on one side of c , and Camille Jordan allowed it even if the function was defined only at c . All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. A real function that

10057-400: The independent variable x always produces an infinitely small change f ( x + α ) − f ( x ) {\displaystyle f(x+\alpha )-f(x)} of the dependent variable y (see e.g. Cours d'Analyse , p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels

10170-413: The infinitesimal definition used today (see microcontinuity ). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c , but Édouard Goursat allowed

10283-422: The interval x 0 − δ < x < x 0 + δ {\displaystyle x_{0}-\delta <x<x_{0}+\delta } be entirely within the domain D {\displaystyle D} , but Jordan removed that restriction. In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of

10396-483: The interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} (the whole real line ) is often called simply a continuous function; one also says that such a function is continuous everywhere . For example, all polynomial functions are continuous everywhere. A function

10509-471: The list elements are positive numbers) provides the harmonic mean. A type of average used in finance is the average percentage return. It is an example of a geometric mean. When the returns are annual, it is called the Compound Annual Growth Rate (CAGR). For example, if we are considering a period of two years, and the investment return in the first year is −10% and the return in the second year

10622-462: The list. Most types of average, however, satisfy permutation -insensitivity: all items count equally in determining their average value and their positions in the list are irrelevant; the average of (1, 2, 3, 4, 6) is the same as that of (3, 2, 6, 4, 1). The arithmetic mean , the geometric mean and the harmonic mean are known collectively as the Pythagorean means . The mode , the median , and

10735-519: The mean for reducing observation errors was indeed mainly developed in astronomy. A possible precursor to the arithmetic mean is the mid-range (the mean of the two extreme values), used for example in Arabian astronomy of the ninth to eleventh centuries, but also in metallurgy and navigation. However, there are various older vague references to the use of the arithmetic mean (which are not as clear, but might reasonably have to do with our modern definition of

10848-427: The mean). In a text from the 4th century, it was written that (text in square brackets is a possible missing text that might clarify the meaning): Even older potential references exist. There are records that from about 700 BC, merchants and shippers agreed that damage to the cargo and ship (their "contribution" in case of damage by the sea) should be shared equally among themselves. This might have been calculated using

10961-512: The meaning in English began with the practice in later medieval and early modern Western merchant-marine law contracts under which if the ship met a bad storm and some of the goods had to be thrown overboard to make the ship lighter and safer, then all merchants whose goods were on the ship were to suffer proportionately (and not whoever's goods were thrown overboard); and more generally there was to be proportionate distribution of any avaria . From there

11074-420: The most general continuous functions, and their definition is the basis of topology . A stronger form of continuity is uniform continuity . In order theory , especially in domain theory , a related concept of continuity is Scott continuity . As an example, the function H ( t ) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M ( t ) denoting

11187-463: The order ≤ {\displaystyle \leq } in the definition of monotonicity is replaced by the strict order < {\displaystyle <} , one obtains a stronger requirement. A function with this property is called strictly increasing (also increasing ). Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing (also decreasing ). A function with either property

11300-406: The order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing or non-increasing ) if, whenever x ≤ y {\displaystyle x\leq y} , then f ( x ) ≥ f ( y ) {\displaystyle f\!\left(x\right)\geq f\!\left(y\right)} , so it reverses the order (see Figure 2). If

11413-435: The owner can claim a proportional contribution from all the parties to the marine venture. The type of calculations used in adjusting general average gave rise to the use of "average" to mean "arithmetic mean". A second English usage, documented as early as 1674 and sometimes spelled "averish", is as the residue and second growth of field crops, which were considered suited to consumption by draught animals ("avers"). There

11526-415: The point x 0 {\displaystyle x_{0}} when the following holds: For any positive real number ε > 0 , {\displaystyle \varepsilon >0,} however small, there exists some positive real number δ > 0 {\displaystyle \delta >0} such that for all x {\displaystyle x} in

11639-511: The reason why monotonic functions are useful in technical work in analysis . Other important properties of these functions include: An important application of monotonic functions is in probability theory . If X {\displaystyle X} is a random variable , its cumulative distribution function F X ( x ) = Prob ( X ≤ x ) {\displaystyle F_{X}\!\left(x\right)={\text{Prob}}\!\left(X\leq x\right)}

11752-467: The remainder. We can formalize this to a definition of continuity. A function C : [ 0 , ∞ ) → [ 0 , ∞ ] {\displaystyle C:[0,\infty )\to [0,\infty ]} is called a control function if A function f : D → R {\displaystyle f:D\to R} is C -continuous at x 0 {\displaystyle x_{0}} if there exists such

11865-428: The requirement that c is in the domain of f ). Second, the limit of that equation has to exist. Third, the value of this limit must equal f ( c ) . {\displaystyle f(c).} (Here, we have assumed that the domain of f does not have any isolated points .) A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c . Intuitively,

11978-466: The right). In the same way, it can be shown that the reciprocal of a continuous function r = 1 / f {\displaystyle r=1/f} (defined by r ( x ) = 1 / f ( x ) {\displaystyle r(x)=1/f(x)} for all x ∈ D {\displaystyle x\in D} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} )

12091-473: The set R {\displaystyle \mathbb {R} } of real numbers. This subset D {\displaystyle D} is the domain of f . Some possible choices include In the case of the domain D {\displaystyle D} being defined as an open interval, a {\displaystyle a} and b {\displaystyle b} do not belong to D {\displaystyle D} , and

12204-1107: The sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition . Given two continuous functions g : D g ⊆ R → R g ⊆ R  and  f : D f ⊆ R → R f ⊆ D g , {\displaystyle g:D_{g}\subseteq \mathbb {R} \to R_{g}\subseteq \mathbb {R} \quad {\text{ and }}\quad f:D_{f}\subseteq \mathbb {R} \to R_{f}\subseteq D_{g},} their composition, denoted as c = g ∘ f : D f → R , {\displaystyle c=g\circ f:D_{f}\to \mathbb {R} ,} and defined by c ( x ) = g ( f ( x ) ) , {\displaystyle c(x)=g(f(x)),}

12317-402: The sum of the weights is not 1.0 (so the output series is a scaled version of the averages). The reason for this is that the analyst is usually interested only in the trend or the periodic behavior. The first recorded time that the arithmetic mean was extended from 2 to n cases for the use of estimation was in the sixteenth century. From the late sixteenth century onwards, it gradually became

12430-442: The terms weakly monotone , weakly increasing and weakly decreasing are often used to refer to non-strict monotonicity. The terms "non-decreasing" and "non-increasing" should not be confused with the (much weaker) negative qualifications "not decreasing" and "not increasing". For example, the non-monotonic function shown in figure 3 first falls, then rises, then falls again. It is therefore not decreasing and not increasing, but it

12543-418: The value G ( 0 ) {\displaystyle G(0)} to be 1, which is the limit of G ( x ) , {\displaystyle G(x),} when x approaches 0, i.e., G ( 0 ) = lim x → 0 sin ⁡ x x = 1. {\displaystyle G(0)=\lim _{x\to 0}{\frac {\sin x}{x}}=1.} Thus, by setting

12656-413: The values of f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} do not matter for continuity on D {\displaystyle D} . The function f is continuous at some point c of its domain if the limit of f ( x ) , {\displaystyle f(x),} as x approaches c through

12769-416: The word was adopted by British insurers, creditors, and merchants for talking about their losses as being spread across their whole portfolio of assets and having a mean proportion. Today's meaning developed out of that, and started in the mid-18th century, and started in English. Marine damage is either particular average , which is borne only by the owner of the damaged property, or general average , where

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