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i فارسی հայերեն italiano lingua franca nova português српски srpski 中文 edit links article talk english read edit view history tools tools move to sidebar hide actions read edit view history general what links here related changes upload file permanent link page information cite this page get shortened url switch to legacy parser print export download as pdf printable version in other projects wikidata item appearance move to sidebar hide from wikipedia the free encyclopedia mathematical function function x f x history of the function concept types by domain and codomain x 𝔹 𝔹 x 𝔹 n x x ℤ ℤ x x ℝ ℝ x ℝ n x x ℂ ℂ x ℂ n x classes properties constant identity linear polynomial rational algebraic analytic smooth continuous measurable injective surjective bijective constructions restriction composition λ inverse generalizations relation binary relation set valued multivalued partial implicit space higher order morphism functor list of specific functions v t e in mathematics a function of a real variable is a function whose domain is a subset of r displaystyle mathbb r many real functions that are often encountered have their domain contain an interval of non empty interior and may be continuous or have some degree of smoothness over one or more intervals each of non empty interior in the domain in older texts the theory of functions of a real variable is often synonymous with what is usually now called real analysis the most widely considered such functions are the real functions which are the real valued functions of a real variable that is the functions of a real variable whose codomain is the set of real numbers nevertheless the codomain of a function of a real variable may be any set however it is often assumed to have a structure of r displaystyle mathbb r vector space over the reals that is the codomain may be a euclidean space a coordinate vector the set of matrices of real numbers of a given size or an r displaystyle mathbb r algebra such as the complex numbers or the quaternions the structure r displaystyle mathbb r vector space of the codomain induces a structure of r displaystyle mathbb r vector space on the functions if the codomain has a structure of r displaystyle mathbb r algebra the same is true for the functions the image of a function of a real variable is a curve in the codomain in this context a function that defines curve is called a parametric equation of the curve when the codomain of a function of a real variable is a finite dimensional vector space the function may be viewed as a sequence of real functions this is often used in applications real function edit the graph of a real function a real function is a function from a subset of r displaystyle mathbb r to r displaystyle mathbb r where r displaystyle mathbb r denotes as usual the set of real numbers that is the domain of a real function is a subset r displaystyle mathbb r and its codomain is r displaystyle mathbb r it is generally assumed that the domain contains an interval of positive length basic examples edit for many commonly used real functions the domain is the whole set of real numbers and the function is continuous and differentiable at every point of the domain one says that these functions are defined continuous and differentiable everywhere this is the case of all polynomial functions including constant functions and linear functions sine and cosine functions exponential function some functions are defined everywhere but not continuous at some points for example the heaviside step function is defined everywhere but not continuous at zero some functions are defined and continuous everywhere but not everywhere differentiable for example the absolute value is defined and continuous everywhere and is differentiable everywhere except for zero the cubic root is defined and continuous everywhere and is differentiable everywhere except for zero many common functions are not defined everywhere but are continuous and differentiable everywhere where they are defined for example a rational function is a quotient of two polynomial functions and is not defined at the zeros of the denominator the tangent function is not defined for π 2 k π displaystyle frac pi 2 k pi where k is any integer the logarithm function is defined only for positive values of the variable some functions are continuous in their whole domain and not differentiable at some points this is the case of the square root is defined only for nonnegative values of the variable and not differentiable at 0 it is differentiable for all positive values of the variable general definition edit a real valued function of a real variable is a function that takes as input a real number commonly represented by the variable x for producing another real number the value of the function commonly denoted f x for simplicity in this article a real valued function of a real variable will be simply called a function to avoid any ambiguity the other types of functions that may occur will be explicitly specified some functions are defined for all real values of the variables one says that they are everywhere defined but some other functions are defined only if the value of the variable is taken in a subset x of r displaystyle mathbb r the domain of the function which is always supposed to contain an interval of positive length in other words a real valued function of a real variable is a function f x r displaystyle f x to mathbb r such that its domain x is a subset of r displaystyle mathbb r that contains an interval of positive length a simple example of a function in one variable could be f x r displaystyle f x to mathbb r x x r x 0 displaystyle x x in mathbb r x geq 0 f x x displaystyle f x sqrt x which is the square root of x image edit main article image mathematics the image of a function f x displaystyle f x is the set of all values of f when the variable x runs in the whole domain of f for a continuous see below for a definition real valued function with a connected domain the image is either an interval or a single value in the latter case the function is a constant function the preimage of a given real number y is the set of the solutions of the equation y f x domain edit the domain of a function of several real variables is a subset of r displaystyle mathbb r that is sometimes explicitly defined in fact if one restricts the domain x of a function f to a subset y x one gets formally a different function the restriction of f to y which is denoted f y in practice it is often not harmful to identify f and f y and to omit the subscript y conversely it is sometimes possible to enlarge naturally the domain of a given function for example by continuity or by analytic continuation this means that it is not worthy to explicitly define the domain of a function of a real variable algebraic structure edit the arithmetic operations may be applied to the functions in the following way for every real number r the constant function x r displaystyle x mapsto r is everywhere defined for every real number r and every function f the function r f x r f x displaystyle rf x mapsto rf x has the same domain as f or is everywhere defined if r 0 if f and g are two functions of respective domains x and y such that x y contains an open subset of r displaystyle mathbb r then f g x f x g x displaystyle f g x mapsto f x g x and f g x f x g x displaystyle f g x mapsto f x g x are functions that have a domain containing x y it follows that the functions of n variables that are everywhere defined and the functions of n variables that are defined in some neighbourhood of a given point both form commutative algebras over the reals r displaystyle mathbb r algebras one may similarly define 1 f x 1 f x displaystyle 1 f x mapsto 1 f x which is a function only if the set of the points x in the domain of f such that f x 0 contains an open subset of r displaystyle mathbb r this constraint implies that the above two algebras are not fields continuity and limit edit limit of a real function of a real variable until the second part of 19th century only continuous functions were considered by mathematicians at that time the notion of continuity was elaborated for the functions of one or several real variables a rather long time before the formal definition of a topological space and a continuous map between topological spaces as continuous functions of a real variable are ubiquitous in mathematics it is worth defining this notion without reference to the general notion of continuous maps between topological space for defining the continuity it is useful to consider the distance function of r displaystyle mathbb r which is an everywhere defined function of 2 real variables d x y x y displaystyle d x y x y a function f is continuous at a point a displaystyle a which is interior to its domain if for every positive real number ε there is a positive real number δ such that f x f a ε displaystyle f x f a varepsilon for all x displaystyle x such that d x a δ displaystyle d x a delta in other words δ may be chosen small enough for having the image by f of the interval of radius δ centered at a displaystyle a contained in the interval of length 2 ε centered at f a displaystyle f a a function is continuous if it is continuous at every point of its domain the limit of a real valued function of a real variable is as follows 1 let a be a point in topological closure of the domain x of the function f the function f has a limit l when x tends toward a denoted l lim x a f x displaystyle l lim _ x to a f x if the following condition is satisfied for every positive real number ε 0 there is a positive real number δ 0 such that f x l ε displaystyle f x l varepsilon for all x in the domain such that d x a δ displaystyle d x a delta if the limit exists it is unique if a is in the interior of the domain the limit exists if and only if the function is continuous at a in this case we have f a lim x a f x displaystyle f a lim _ x to a f x when a is in the boundary of the domain of f and if f has a limit at a the latter formula allows to extend by continuity the domain of f to a calculus edit one can collect a number of functions each of a real variable say y 1 f 1 x y 2 f 2 x y n f n x displaystyle y_ 1 f_ 1 x quad y_ 2 f_ 2 x ldots y_ n f_ n x into a vector parametrized by x y y 1 y 2 y n f 1 x f 2 x f n x displaystyle mathbf y y_ 1 y_ 2 ldots y_ n f_ 1 x f_ 2 x ldots f_ n x the derivative of the vector y is the vector derivatives of f i x for i 1 2 n d y d x d y 1 d x d y 2 d x d y n d x displaystyle frac d mathbf y dx left frac dy_ 1 dx frac dy_ 2 dx ldots frac dy_ n dx right one can also perform line integrals along a space curve parametrized by x with position vector r r x by integrating with respect to the variable x a b y x d r a b y x d r x d x d x displaystyle int _ a b mathbf y x cdot d mathbf r int _ a b mathbf y x cdot frac d mathbf r x dx dx where is the dot product and x a and x b are the start and endpoints of the curve theorems edit with the definitions of integration and derivatives key theorems can be formulated including the fundamental theorem of calculus integration by parts and taylor s theorem evaluating a mixture of integrals and derivatives can be done by using theorem differentiation under the integral sign implicit functions edit a real valued implicit function of a real variable is not written in the form y f x instead the mapping is from the space r displaystyle mathbb r 2 to the zero element in r displaystyle mathbb r just the ordinary zero 0 ϕ r 2 0 displaystyle phi mathbb r 2 to 0 and ϕ x y 0 displaystyle phi x y 0 is an equation in the variables implicit functions are a more general way to represent functions since if y f x displaystyle y f x then we can always define ϕ x y y f x 0 displaystyle phi x y y f x 0 but the converse is not always possible i e not all implicit functions have the form of this equation one dimensional space curves in r displaystyle mathbb r n edit space curve in 3d the position vector r is parametrized by a scalar t at r a the red line is the tangent to the curve and the blue plane is normal to the curve formulation edit given the functions r 1 r 1 t r 2 r 2 t r n r n t all of a common variable t so that r 1 r r r 2 r r r n r r r 1 r 1 t r 2 r 2 t r n r n t displaystyle begin aligned r_ 1 mathbb r rightarrow mathbb r quad r_ 2 mathbb r rightarrow mathbb r cdots quad r_ n mathbb r rightarrow mathbb r r_ 1 r_ 1 t quad r_ 2 r_ 2 t cdots quad r_ n r_ n t end aligned or taken together r r r n r r t displaystyle mathbf r mathbb r rightarrow mathbb r n quad mathbf r mathbf r t then the parametrized n tuple r t r 1 t r 2 t r n t displaystyle mathbf r t r_ 1 t r_ 2 t ldots r_ n t describes a one dimensional space curve tangent line to curve edit at a point r t c a a 1 a 2 a n for some constant t c the equations of the one dimensional tangent line to the curve at that point are given in terms of the ordinary derivatives of r 1 t r 2 t r n t and r with respect to t r 1 t a 1 d r 1 t d t r 2 t a 2 d r 2 t d t r n t a n d r n t d t displaystyle frac r_ 1 t a_ 1 dr_ 1 t dt frac r_ 2 t a_ 2 dr_ 2 t dt cdots frac r_ n t a_ n dr_ n t dt normal plane to curve edit the equation of the n dimensional hyperplane normal to the tangent line at r a is p 1 a 1 d r 1 t d t p 2 a 2 d r 2 t d t p n a n d r n t d t 0 displaystyle p_ 1 a_ 1 frac dr_ 1 t dt p_ 2 a_ 2 frac dr_ 2 t dt cdots p_ n a_ n frac dr_ n t dt 0 or in terms of the dot product p a d r t d t 0 displaystyle mathbf p mathbf a cdot frac d mathbf r t dt 0 where p p 1 p 2 p n are points in the plane not on the space curve relation to kinematics edit kinematic quantities of a classical particle mass m position r velocity v acceleration a the physical and geometric interpretation of d r t dt is the velocity of a point like particle moving along the path r t treating r as the spatial position vector coordinates parametrized by time t and is a vector tangent to the space curve for all t in the instantaneous direction of motion at t c the space curve has a tangent vector d r t dt t c and the hyperplane normal to the space curve at t c is also normal to the tangent at t c any vector in this plane p a must be normal to d r t dt t c similarly d 2 r t dt 2 is the acceleration of the particle and is a vector normal to the curve directed along the radius of curvature matrix valued functions edit a matrix can also be a function of a single variable for example the rotation matrix in 2d r θ cos θ sin θ sin θ cos θ displaystyle r theta begin bmatrix cos theta sin theta sin theta cos theta end bmatrix is a matrix valued function of rotation angle of about the origin similarly in special relativity the lorentz transformation matrix for a pure boost without rotations λ β 1 1 β 2 β 1 β 2 0 0 β 1 β 2 1 1 β 2 0 0 0 0 1 0 0 0 0 1 displaystyle lambda beta begin bmatrix frac 1 sqrt 1 beta 2 frac beta sqrt 1 beta 2 0 0 frac beta sqrt 1 beta 2 frac 1 sqrt 1 beta 2 0 0 0 0 1 0 0 0 0 1 end bmatrix ...
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