Real-valued function

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 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.

Algebraic structure

Let be the set of all functions from a set 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:

These operations extend to partial functions from to with the restriction that the partial functions and are defined only if the domains of and have a nonempty intersection; in this case, their domain is the intersection of the domains of and.
Also, since is an ordered set, there is a partial order
on which makes a partially ordered ring.

Measurable

The σ-algebra of Borel sets is an important structure on real numbers. If has its σ-algebra and a function is such that the preimage of any Borel set belongs to that σ-algebra, then is said to be measurable. Measurable functions also form a vector space and an algebra as explained above in.
Moreover, a set of real-valued functions on can actually define a σ-algebra on generated by all preimages of all Borel sets. This is the way how σ-algebras arise in probability theory, where real-valued functions on the sample space are real-valued random variables.

Continuous

Real numbers form a topological space and a complete metric space. Continuous real-valued functions 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.
The concept of metric space itself is defined with a real-valued function of two variables, the metric, which is continuous. The space of continuous functions on a compact Hausdorff space has a particular importance. Convergent sequences also can be considered as real-valued continuous functions on a special topological space.
Continuous functions also form a vector space and an algebra as explained above in, and are a subclass of [|measurable functions] because any topological space has the σ-algebra generated by open sets.

Smooth

Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be the real coordinate space, a topological vector space, an open subset of them, or a smooth manifold.
Spaces of smooth functions also are vector spaces and algebras as explained above in and are subspaces of the space of [|continuous functions].

Appearances in measure theory

A measure on a set is a non-negative real-valued functional on a σ-algebra of subsets. L^p spaces on sets with a measure are defined from aforementioned [|real-valued measurable functions], although they are actually quotient spaces. More precisely, whereas a function satisfying an appropriate summability condition defines an element of L^p space, in the opposite direction for any and which is not an atom, the value is undefined. Though, real-valued L^p spaces still have some of the structure described above in. Each of L^p spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes, namely
For example, pointwise product of two L² functions belongs to L¹.

Other appearances

Other contexts where real-valued functions and their special properties are used include monotonic functions, convex functions, harmonic and subharmonic functions, analytic functions, algebraic functions, and polynomials.