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In mathematics, areal-valued function is afunction whosevalues arereal numbers. In other words, it is a function that assigns a real number to each member of itsdomain.

Real-valuedfunctions of a real variable (commonly calledreal functions) and real-valuedfunctions of several real variables are the main object of study ofcalculus and, more generally,real analysis. In particular, manyfunction spaces consist of real-valued functions.

Let${\displaystyle \mathcal{F}(X,\mathbb{R})}$ be the set of all functions from asetX to real numbers${\displaystyle \mathbb{R}}$. Because${\displaystyle \mathbb{R}}$ is afield,${\displaystyle \mathcal{F}(X,\mathbb{R})}$ may be turned into avector space and acommutative algebra over the reals with the following operations:

• ${\displaystyle f+g:x\mapsto f(x)+g(x)}$ –vector addition
• ${\displaystyle \mathbf{0}:x\mapsto 0}$ –additive identity
• ${\displaystyle cf:x\mapsto cf(x),c\in \mathbb{R}}$ –scalar multiplication
• ${\displaystyle fg:x\mapsto f(x)g(x)}$ –pointwise multiplication

These operations extend topartial functions fromX to${\displaystyle \mathbb{R},}$ with the restriction that the partial functionsf +g andfg are defined only if thedomains off andg have a nonempty intersection; in this case, their domain is the intersection of the domains off andg.

Also, since${\displaystyle \mathbb{R}}$ is an ordered set, there is apartial order

• ${\displaystyle f\le g⟺\mathrm{\forall }x:f(x)\le g(x),}$

on${\displaystyle \mathcal{F}(X,\mathbb{R}),}$ which makes${\displaystyle \mathcal{F}(X,\mathbb{R})}$ apartially ordered ring.

Theσ-algebra ofBorel sets is an important structure on real numbers. IfX has its σ-algebra and a functionf is such that thepreimagef ⁻¹(B) of any Borel setB belongs to that σ-algebra, thenf is said to bemeasurable. Measurable functions also form a vector space and an algebra as explained above in§ Algebraic structure.

Moreover, a set (family) of real-valued functions onX can actuallydefine a σ-algebra onX generated by all preimages of all Borel sets (or ofintervals only, it is not important). This is the way how σ-algebras arise in (Kolmogorov's)probability theory, where real-valued functions on thesample spaceΩ are real-valuedrandom variables.

Real numbers form atopological space and acomplete metric space.Continuous real-valued functions (which implies thatX is a topological space) are important in theoriesof topological spaces andof metric spaces. Theextreme value theorem states that for any real continuous function on acompact space its globalmaximum and minimum exist.

The concept ofmetric space itself is defined with a real-valued function of two variables, themetric, which is continuous. The space ofcontinuous 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 ofmeasurable functions because any topological space has the σ-algebra generated by open (or closed) sets.

Real numbers are used as the codomain to define smooth functions. A domain of a real smooth function can be thereal coordinate space (which yields areal multivariable function), atopological vector space,^[1] anopen subset of them, or asmooth manifold.

Spaces of smooth functions also are vector spaces and algebras as explained above in§ Algebraic structure and are subspaces of the space ofcontinuous functions.

Ameasure on a set is anon-negative real-valued functional on a σ-algebra of subsets.^[2]L^p spaces on sets with a measure are defined from aforementionedreal-valued measurable functions, although they are actuallyquotient spaces. More precisely, whereas a function satisfying an appropriatesummability condition defines an element of L^p space, in the opposite direction for anyf ∈ L^p(X) andx ∈X which is not anatom, the valuef(x) isundefined. Though, real-valued L^p spaces still have some of the structure described above in§ Algebraic structure. Each of L^p spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changesp, namely

${\displaystyle \cdot :L^{1/\alpha }\times L^{1/\beta }\to L^{1/(\alpha +\beta )},0\le \alpha ,\beta \le 1,\alpha +\beta \le 1.}$

For example, pointwise product of two L² functions belongs to L¹.

Other contexts where real-valued functions and their special properties are used includemonotonic functions (onordered sets),convex functions (on vector andaffine spaces),harmonic andsubharmonic functions (onRiemannian manifolds),analytic functions (usually of one or more real variables),algebraic functions (on realalgebraic varieties), andpolynomials (of one or more real variables).

• Real analysis
• Partial differential equations, a major user of real-valued functions
• Norm (mathematics)
• Scalar (mathematics)

• Apostol, Tom M. (1974).Mathematical Analysis (2nd ed.). Addison–Wesley.ISBN 978-0-201-00288-1.
• Gerald Folland, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999,ISBN 0-471-31716-0.
• Rudin, Walter (1976).Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill.ISBN 978-0-07-054235-8.

Weisstein, Eric W."Real Function".MathWorld.

• Mathematical analysis
• Types of functions
• General topology
• Metric geometry
• Vector spaces
• Measure theory

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