Real-valued function

In mathematics, a real-valued function is a function whose

values

are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

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

and, more generally,

real analysis

. In particular, many function spaces consist of real-valued functions.

Algebraic structure

Let $(X,)$ be the set of all functions from a set to real numbers $\mathbb R$. Because $\mathbb R$ is a field, $(X,)$ may be turned into a vector space and a commutative algebra over the reals with the following operations:
*$f+g: x \mapsto f(x) + g(x)$ –

vector addition

*$\mathbf: x \mapsto 0$ – additive identity
*$c f: x \mapsto c f(x),\quad c \in \mathbb R$ –

scalar multiplication

*$f g: x \mapsto f(x)g(x)$ – pointwise multiplication

These operations extend to

partial function

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
*$\ f \le g \quad\iff\quad \forall x: f(x) \le g(x),$
on $(X,),$ which makes $(X,)$ a partially ordered ring.

Measurable

The σ-algebra of Borel sets 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, where real-valued functions on the

sample space

are real-valued random variables.

Continuous

Real numbers form a topological space and a complete metric space. Continuous 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 states that for any real continuous function on a compact space its global maximum and minimum exist.

The concept of metric space itself is defined with a real-valued function of two variables, the ''metric'', 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 (which yields a real multivariable function), 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, 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 function

s (on vector and affine spaces), harmonic and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one or more real variables), algebraic functions (on real algebraic varieties), and

polynomial

s (of one or more real variables).

See also

*

Real analysis

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

Footnotes

References

*
* Gerald Folland, 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