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.
Algebraic structure Let F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} be the set of all functions from a set X to real numbers R {\displaystyle \mathbb {R} } . Because R {\displaystyle \mathbb {R} } is a field, F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} may be turned into a vector space and a commutative algebra over the reals with the following operations:
f + g : x ↦ f ( x ) + g ( x ) {\displaystyle f+g:x\mapsto f(x)+g(x)} – vector addition
0 : x ↦ 0 {\displaystyle \mathbf {0} :x\mapsto 0} – additive identity
c f : x ↦ c f ( x ) , c ∈ R {\displaystyle cf:x\mapsto cf(x),\quad c\in \mathbb {R} } – scalar multiplication
f g : x ↦ f ( x ) g ( x ) {\displaystyle fg:x\mapsto f(x)g(x)} – pointwise multiplication These operations extend to partial functions from X to R , {\displaystyle \mathbb {R} ,} 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 R {\displaystyle \mathbb {R} } is an ordered set, there is a partial order
f ≤ g ⟺ ∀ x : f ( x ) ≤ g ( x ) , {\displaystyle \ f\leq g\quad \iff \quad \forall x:f(x)\leq g(x),}
on F ( X , R ) , {\displaystyle {\mathcal {F}}(X,{\mathbb {R} }),} which makes F ( X , R ) {\displaystyle {\mathcal {F}}(X,{\mathbb {R} })} a partially ordered ring.
Measurable
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 preimage 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 § Algebraic structure. 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.
Continuous 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. 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 § Algebraic structure, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) 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 (which yields a real multivariable function), 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 § Algebraic structure 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. Lp 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 Lp space, in the opposite direction for any f ∈ Lp(X) and x ∈ X which is not an atom, the value f(x) is undefined. Though, real-valued Lp spaces still have some of the structure described above in § Algebraic structure. Each of Lp spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes p, namely
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