TheInfoList

In mathematics, a real-valued function is a
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 $\left(X,\right)$ be the set of all functions from a set to real numbers $\mathbb R$. Because $\mathbb R$ is 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 area of mowed grass * Meadow, a grassl ...
, $\left(X,\right)$ 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: *$f+g: x \mapsto f\left(x\right) + g\left(x\right)$
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 ...

*$\mathbf: x \mapsto 0$
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 ...
*$c f: x \mapsto c f\left(x\right),\quad c \in \mathbb R$
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 ...

*$f g: x \mapsto f\left(x\right)g\left(x\right)$
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 $\mathbb R,$ 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 $\mathbb R$ 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 ...
*$\ f \le g \quad\iff\quad \forall x: f\left(x\right) \le g\left(x\right),$ on $\left(X,\right),$ which makes $\left(X,\right)$ 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 of
Borel 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 a
topological 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 the
real 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 a
non-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 ...
.
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 defines an element of L''p'' space, in the opposite direction for any and which is not an
atom 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 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 :$\sdot: L^ \times L^ \to L^,\quad 0 \le \alpha,\beta \le 1,\quad\alpha+\beta \le 1.$ 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 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).

*
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 ...

# 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, . *