View source for Real-valued function

{{short description|Mathematical function that takes real values}}
{{refimprove|date=June 2013}}
[[Image:Weights 20mg~500g.jpg|thumb|right|[[Mass]] measured in [[gram]]s is a function from this collection of weight to [[positive number|positive]] real numbers. The term "[[weight function]]", an allusion to this example, is used in pure and applied mathematics.]]
{{Functions}}
In mathematics, a '''real-valued function''' is a [[function (mathematics)|function]] whose [[codomain|values]] are [[real number]]s.  In other words, it is a function that assigns a real number to each member of its [[domain of a function|domain]].

Real-valued [[function of a real variable|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 space]]s consist of real-valued functions.

== Algebraic structure ==
Let <math>{\mathcal F}(X,{\mathbb R})</math> be the set of all functions from a [[set (mathematics)|set]] {{mvar|X}} to real numbers <math>\mathbb R</math>. Because <math>\mathbb R</math> is a [[field (mathematics)|field]], <math>{\mathcal F}(X,{\mathbb R})</math> may be turned into a [[vector space]] and a [[commutative algebra (structure)|commutative algebra]] over the reals with the following operations:
*<math>f+g: x \mapsto f(x) + g(x)</math> – [[vector addition]]
*<math>\mathbf{0}: x \mapsto 0</math> – [[additive identity]]
*<math>c f: x \mapsto c f(x),\quad c \in \mathbb R</math> – [[scalar multiplication]]
*<math>f g: x \mapsto f(x)g(x)</math> – [[pointwise]] multiplication

These operations extend to [[partial function]]s from {{mvar|X}} to <math>\mathbb R,</math> with the restriction that the partial functions {{math|''f'' + ''g''}} and {{math|''f'' ''g''}} are defined only if the [[domain of a function|domains]] of {{mvar|f}} and {{mvar|g}} have a nonempty intersection; in this case, their domain is the intersection of the domains of {{mvar|f}} and {{mvar|g}}.

Also, since <math>\mathbb R</math> is an ordered set, there is a [[partial order]]
*<math>\ f \le g \quad\iff\quad \forall x: f(x) \le g(x),</math>
on <math>{\mathcal F}(X,{\mathbb R}),</math> which makes <math>{\mathcal F}(X,{\mathbb R}) </math> a [[partially ordered ring]].

== Measurable ==
{{see also|Borel function}}
The [[σ-algebra]] of [[Borel set]]s is an important structure on real numbers. If {{mvar|X}} has its σ-algebra and a function {{mvar|f}} is such that the [[preimage]] {{math|''f'' <sup>−1</sup>(''B'')}} of any Borel set {{mvar|''B''}} belongs to that σ-algebra, then {{mvar|f}} is said to be [[measurable function|measurable]]. Measurable functions also form a vector space and an algebra as explained above in {{slink||Algebraic structure}}.

Moreover, a set (family) of real-valued functions on {{mvar|X}} can actually ''define'' a σ-algebra on {{mvar|X}} generated by all preimages of all Borel sets (or of [[interval (mathematics)|intervals]] only, it is not important). This is the way how σ-algebras arise in ([[Kolmogorov's axioms|Kolmogorov's]]) [[probability theory]], where real-valued functions on the [[sample space]] {{math|&Omega;}} are real-valued [[random variable]]s.

== Continuous ==
Real numbers form a [[topological space]] and a [[complete metric space]]. [[Continuous function|Continuous]] real-valued functions (which implies that {{mvar|X}} is a topological space) are important in theories [[general topology|of topological spaces]] and [[metric geometry|of metric spaces]]. The [[extreme value theorem]] states that for any real continuous function on a [[compact space]] its global [[maxima and minima|maximum and minimum]] exist.

The concept of [[metric space]] itself is defined with a real-valued function of two variables, the ''[[metric (mathematics)|metric]]'', which is continuous. The space of [[continuous functions on a compact Hausdorff space]] has a particular importance. [[Convergent sequence]]s 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 {{slink||Algebraic structure}}, and are a subclass of [[#Measurable|measurable functions]] because any topological space has the σ-algebra generated by open (or closed) sets.

== Smooth ==
{{main|Smooth function}}
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]],<ref>Different definitions of [[derivative]] exist in general, but for finite [[dimension (vector space)|dimensions]] they result in equivalent definitions of classes of smooth functions.</ref> an [[open subset]] of them, or a [[smooth manifold]].

Spaces of smooth functions also are vector spaces and algebras as explained above in {{slink||Algebraic structure}} and are subspaces of the space of [[#Continuous|continuous functions]].

== Appearances in measure theory ==
A [[measure (mathematics)|measure]] on a set is a [[non-negative]] real-valued functional on a σ-algebra of subsets.<ref>Actually, a measure may have values in {{closed-closed|0, +∞}}: see [[extended real number line]].</ref> [[Lp space|L<sup>''p''</sup> spaces]] on sets with a measure are defined from aforementioned [[#Measurable|real-valued measurable functions]], although they are actually [[Quotient space (topology)|quotient space]]s. More precisely, whereas a function satisfying an appropriate [[integral|summability condition]] defines an element of L<sup>''p''</sup> space, in the opposite direction for any {{math|''f'' ∈ L<sup>''p''</sup>(''X'')}} and {{math|''x'' ∈ ''X''}} which is not an [[atom (measure theory)|atom]], the value {{math|''f''(''x'')}} is [[well-definition|undefined]]. Though, real-valued L<sup>''p''</sup> spaces still have some of the structure described above in {{slink||Algebraic structure}}. Each of L<sup>''p''</sup> spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes {{mvar|p}}, namely
:<math>\sdot: L^{1/\alpha} \times L^{1/\beta} \to L^{1/(\alpha+\beta)},\quad
0 \le \alpha,\beta \le 1,\quad\alpha+\beta \le 1.</math>
For example, pointwise product of two L<sup>2</sup> functions belongs to L<sup>1</sup>.

== Other appearances ==
Other contexts where real-valued functions and their special properties are used include [[monotonic function]]s (on [[ordered set]]s), [[convex function]]s (on vector and [[affine space]]s), [[harmonic function|harmonic]] and [[subharmonic function|subharmonic]] functions (on [[Riemannian manifold]]s), [[analytic function]]s (usually of one or more real variables), [[algebraic function]]s (on real [[algebraic variety|algebraic varieties]]), and [[polynomial]]s (of one or more real variables).

==See also==
* [[Real analysis]]
* [[Partial differential equation]]s, a major user of real-valued functions
* [[Norm (mathematics)]]
* [[Scalar (mathematics)]]

==Footnotes==
{{reflist}}

==References==
* {{cite book|last=Apostol|first=Tom M.|date=1974|title=Mathematical Analysis|edition=2nd|publisher=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}}.
* {{Cite book|title=Principles of Mathematical Analysis|last=Rudin|first=Walter|publisher=McGraw-Hill|year=1976|isbn=978-0-07-054235-8|edition=3rd|location=New York}}

==External links==
{{MathWorld |title=Real Function |id=RealFunction}}

[[Category:Mathematical analysis]]
[[Category:Types of functions]]
[[Category:General topology]]
[[Category:Metric geometry]]
[[Category:Vector spaces]]
[[Category:Measure theory]]