In mathematics, a real-valued function is a

^{''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 defines an element of L^{''p''} space, in the opposite direction for any and which is not an ^{''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
:$\backslash sdot:\; L^\; \backslash times\; L^\; \backslash to\; L^,\backslash quad\; 0\; \backslash le\; \backslash alpha,\backslash beta\; \backslash le\; 1,\backslash quad\backslash alpha+\backslash beta\; \backslash le\; 1.$
For example, pointwise product of two L^{2} functions belongs to L^{1}.

function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

whose values
In ethics
Ethics or moral philosophy is a branch of philosophy
Philosophy (from , ) is the study of general and fundamental questions, such as those about Metaphysics, existence, reason, Epistemology, knowledge, Ethics, values, Philoso ...

are real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

s. In other words, it is a function that assigns a real number to each member of its domain
Domain may refer to:
Mathematics
*Domain of a function
In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...

.
Real-valued functions of a real variable (commonly called ''real functions'') and real-valued functions of several real variables
In mathematical analysis its applications, a function of several real variables or real multivariate function is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A comp ...

are the main object of study of calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations ...

and, more generally, real analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

. In particular, many function space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

s consist of real-valued functions.
Algebraic structure

Let $(X,)$ be the set of all functions from a set to real numbers $\backslash mathbb\; R$. Because $\backslash mathbb\; R$ is afield
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, $(X,)$ may be turned into a vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

and a commutative algebra
Commutative algebra is the branch of algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry ...

over the reals with the following operations:
*$f+g:\; x\; \backslash mapsto\; f(x)\; +\; g(x)$ – vector addition
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

*$\backslash mathbf:\; x\; \backslash mapsto\; 0$ – additive identity In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

*$c\; f:\; x\; \backslash mapsto\; c\; f(x),\backslash quad\; c\; \backslash in\; \backslash mathbb\; R$ – scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module (mathematics), module in abstract algebra). In common geometrical contexts, scalar multiplication of a re ...

*$f\; g:\; x\; \backslash mapsto\; f(x)g(x)$ – pointwiseIn mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined o ...

multiplication
These operations extend to partial function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s from to $\backslash mathbb\; R,$ 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 $\backslash mathbb\; R$ is an ordered set, there is a partial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

*$\backslash \; f\; \backslash le\; g\; \backslash quad\backslash iff\backslash quad\; \backslash forall\; x:\; f(x)\; \backslash le\; g(x),$
on $(X,),$ which makes $(X,)$ a partially ordered ring
In abstract algebra, a partially ordered ring is a Ring (mathematics), ring (''A'', +, ·), together with a ''compatible partial order'', i.e. a partial order \leq on the underlying set ''A'' that is compatible with the ring operations in the sense ...

.
Measurable

The σ-algebra ofBorel set In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and the ...

s is an important structure on real numbers. If has its σ-algebra and a function is such that the preimage
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of any Borel set belongs to that σ-algebra, then is said to be measurable
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

. Measurable functions also form a vector space and an algebra as explained above in .
Moreover, a set (family) of real-valued functions on can actually ''define'' a σ-algebra on 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
Probability theory is the branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are containe ...

, where real-valued functions on the sample space
In probability theory
Probability theory is the branch of concerned with . Although there are several different , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of . Typically these axiom ...

are real-valued random variable
A random variable is a variable whose values depend on outcomes of a random
In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no ...

s.
Continuous

Real numbers form atopological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

and a complete metric space
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...

. Continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ga ...

real-valued functions (which implies that is a topological space) are important in theories of topological spaces and of metric spaces. The extreme value theorem
In calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of ...

states that for any real continuous function on a compact space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

its global maximum and minimum exist.
The concept of metric space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

itself is defined with a real-valued function of two variables, the ''metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
Mathematics
* Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space
* Metric tensor, in differential geomet ...

'', which is continuous. The space of continuous functions on a compact Hausdorff spaceIn mathematical analysis, and especially functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, ...

has a particular importance. Convergent sequence
As the positive integer
An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number that can be written without a Fraction (mathematics), fractional component. For example, 21, 4, 0, ...

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 , 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 thereal coordinate space
In mathematics, a real coordinate space of dimension , written ( ) or is a coordinate space over the real numbers. This means that it is the set of the tuple, -tuples of real numbers (sequences of real numbers). With component-wise addition a ...

(which yields a real multivariable function), a topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space (an Abstra ...

, an open subset
Open or OPEN may refer to:
Music
* Open (band)
Open is a band.
Background
Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australian ...

of them, or a smooth manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ...

.
Spaces of smooth functions also are vector spaces and algebras as explained above in and are subspaces of the space of continuous functions
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
Appearances in measure theory

A measure on a set is anon-negative
In mathematics, the sign of a real number is its property of being either positive, negative number, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third ...

real-valued functional on a σ-algebra of subsets.Actually, a measure may have values in : see extended real number line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

. Latom
An atom is the smallest unit of ordinary matter
In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of ato ...

, the value is undefined. Though, real-valued LOther appearances

Other contexts where real-valued functions and their special properties are used includemonotonic function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s (on ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a Set (mathematics), set. A poset consists of a set toget ...

s), convex function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s (on vector and affine space
In mathematics, an affine space is a geometric Structure (mathematics), structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping on ...

s), harmonic
A harmonic is any member of the harmonic series
Harmonic series may refer to either of two related concepts:
*Harmonic series (mathematics)
*Harmonic series (music)
{{Disambig .... The term is employed in various disciplines, including music ...

and subharmonic
In music
Music is the art of arranging sounds in time through the elements of melody, harmony, rhythm, and timbre. It is one of the universal cultural aspects of all human societies. General definitions of music include common elements ...

functions (on Riemannian manifold
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integr ...

s), analytic function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

s (usually of one or more real variables), algebraic functionIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s (on real algebraic varieties
Algebraic varieties are the central objects of study in algebraic geometry
Algebraic geometry is a branch of mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures ...

), and polynomial
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s (of one or more real variables).
See also

*Real analysis
200px, The first four partial sums of the Fourier series for a square wave. Fourier series are an important tool in real analysis.">square_wave.html" ;"title="Fourier series for a square wave">Fourier series for a square wave. Fourier series are a ...

* Partial differential equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a Multivariable calculus, multivariable function.
The function is often thought of as an "unknown" to be sol ...

s, a major user of real-valued functions
* Norm (mathematics)
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

* Scalar (mathematics)
A scalar is an element of a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an ar ...

Footnotes

References

* *Gerald Folland
Gerald Budge Folland is an United States, American mathematician and a professor of mathematics at the University of Washington. His areas of interest are harmonic analysis (on both Euclidean space and Lie groups), differential equations, and math ...

, Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, .
*
External links

{{MathWorld , title=Real Function , id=RealFunction Mathematical analysis Types of functions General topology Metric geometry Vector spaces Measure theory