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R is a programming language for statistical computing and data visualization. It has been adopted in the fields of data mining, bioinformatics and data analysis. [9] The core R language is augmented by a large number of extension packages, containing reusable code, documentation, and sample data. R software is open-source and free software.
See main article at Function space, especially the functional analysis section. Let X be a non-empty arbitrary set and V an arbitrary vector space over F. The space of all functions from X to V is a vector space over F under pointwise addition and multiplication. That is, let f : X → V and g : X → V denote two functions, and let α in F. We ...
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
This function is a test function on and is an element of (). The support of this function is the closed unit disk in . It is non-zero on the open unit disk and it is equal to 0 everywhere outside of it.
R is equivalent to the set of all total computable functions in the sense that: a decision problem is in R if and only if its indicator function is computable, a total function is computable if and only if its graph is in R.
Given its domain and its codomain, a function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function, a popular means of illustrating the function. [ note 1 ] [ 4 ] When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane.
Let F be a field and let X be any set. The functions X → F can be given the structure of a vector space over F where the operations are defined pointwise, that is, for any f, g : X → F, any x in X, and any c in F, define (+) = + () = When the domain X has additional structure, one might consider instead the subset (or subspace) of all such functions which respect that structure.
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 ...