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Concept
Real-valued function
Summary
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.
Official source
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