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In mathematics, a real-valued function is a function whose values are
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
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, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function **Domain of holomorphy of a function * ...
. 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 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 generalizati ...
and, more generally, real analysis. In particular, many
function space In mathematics, a function space is a set of functions between two fixed sets. Often, the domain and/or codomain will have additional structure which is inherited by the function space. For example, the set of functions from any set into a vect ...
s consist of real-valued functions.


Algebraic structure

Let (X,) be the set of all functions from a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
to real numbers \mathbb R. Because \mathbb R is a field, (X,) may be turned into a
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
and a
commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prom ...
over the reals with the following operations: *f+g: x \mapsto f(x) + g(x)vector addition *\mathbf: x \mapsto 0additive identity *c f: x \mapsto c f(x),\quad c \in \mathbb Rscalar multiplication *f g: x \mapsto f(x)g(x)
pointwise In 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 ...
multiplication These operations extend to partial functions 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 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. A poset consists of a set together with a binary ...
*\ f \le g \quad\iff\quad \forall x: f(x) \le g(x), on (X,), which makes (X,) a partially ordered ring.


Measurable

The σ-algebra of
Borel set In mathematics, a Borel set is any set in a topological space that can be formed from open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets are na ...
s is an important structure on real numbers. If has its σ-algebra and a function is such that the preimage of any Borel set belongs to that σ-algebra, then is said to be measurable. 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 concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, where real-valued functions on the sample space are real-valued random variables.


Continuous

Real numbers form a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poin ...
and a complete metric space.
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 g ...
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, the extreme value theorem states that if a real-valued function f is continuous on the closed interval ,b/math>, then f must attain a maximum and a minimum, each at least once. That is, there exist numbers c and d in ,b/math> s ...
states that for any real continuous function on a
compact space In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space by making precise the idea of a space having no "punctures" or "missing endpoints", i ...
its global maximum and minimum exist. The concept of
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...
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 * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathe ...
'', 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 , 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, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector ...
(which yields a
real multivariable function In mathematical analysis and its applications, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of ...
), 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 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.Actually, a measure may have values in : see extended real number line. 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 summability condition defines an element of L''p'' space, in the opposite direction for any and which is not an
atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, a ...
, 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 functions (on ordered sets), convex functions (on vector and affine spaces),
harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...
and subharmonic functions (on Riemannian manifolds),
analytic function In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions. Functions of each type are infinitely differentiable, but complex ...
s (usually of one or more real variables), algebraic functions (on real algebraic varieties), and
polynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exampl ...
s (of one or more real variables).


See also

* Real analysis * Partial differential equations, a major user of real-valued functions *
Norm (mathematics) In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and ...
*
Scalar (mathematics) A scalar is an element of a field which is used to define a ''vector space''. In linear algebra, real numbers or generally elements of a field are called scalars and relate to vectors in an associated vector space through the operation of sca ...


Footnotes


References

* *
Gerald Folland Gerald Budge Folland is an American mathematician and a professor of mathematics at the University of Washington. He is the author of several textbooks on mathematical analysis. His areas of interest include harmonic analysis (on both Euclidean ...
, 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