In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.
Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions.
Let be the set of all functions from a set X to real numbers . Because is a field, may be turned into a vector space and a commutative algebra over the reals with the following operations:
– vector addition
– additive identity
– scalar multiplication
– pointwise multiplication
These operations extend to partial functions from X to with the restriction that the partial functions f + g and f g are defined only if the domains of f and g have a nonempty intersection; in this case, their domain is the intersection of the domains of f and g.
Also, since is an ordered set, there is a partial order
on which makes a partially ordered ring.
Borel function
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 f −1(B) of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a vector space and an algebra as explained above in .
Moreover, a set (family) of real-valued functions on X can actually define a σ-algebra on X generated by all preimages of all Borel sets (or of intervals only, it is not important). This is the way how σ-algebras arise in (Kolmogorov's) probability theory, where real-valued functions on the sample space Ω are real-valued random variables.
Real numbers form a topological space and a complete metric space. Continuous real-valued functions (which implies that X is a topological space) 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.
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.
En son coeur, c'est un cours d'analyse fonctionnelle pour les physiciens et traite les bases de théorie de mesure, des espaces des fonctions et opérateurs linéaires.
In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The "input" variables take real values, while the "output", also called the "value of the function", may be real or complex.
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function.
In mathematics, a limit is the value that a function (or sequence) approaches as the input (or index) approaches some value. Limits are essential to calculus and mathematical analysis, and are used to define continuity, derivatives, and integrals. The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to and direct limit in . In formulas, a limit of a function is usually written as (although a few authors use "Lt" instead of "lim") and is read as "the limit of f of x as x approaches c equals L".
Explores dominant balance analysis in solving the quintic polynomial, revealing insights into root behavior and the importance of symbolic expressions.