World Library  
Flag as Inappropriate
Email this Article

Real-valued function

Article Id: WHEBN0022284647
Reproduction Date:

Title: Real-valued function  
Author: World Heritage Encyclopedia
Language: English
Subject: Froda's theorem, Partial function, Function (mathematics), Limit (mathematics), Analysis
Publisher: World Heritage Encyclopedia

Real-valued function

Mass measured in grams is a function from this collection of weights to positive real numbers. The term "weight function", an allusion to this example, is used in pure and applied mathematics.

In mathematics, a real-valued function or real 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 functions are not especially interesting in general, but many important function spaces are defined to consist of real functions.

In general

Let X be an arbitrary set. Let {\mathcal F}(X,{\mathbb R}) to denote the set of all functions from X to real numbers R. Because R is a field, {\mathcal F}(X,{\mathbb R}) is a vector space and a commutative algebra (structure) over reals:

Also, since R is an ordered set, there is a partial order on {\mathcal F}(X,{\mathbb R}) :

  • \ f \le g \quad\iff\quad \forall x: f(x) \le g(x).

{\mathcal F}(X,{\mathbb R}) is a partially ordered ring.


The σ-algebra of Borel sets is an important structure on real numbers. If X has its σ-algebra and a function f is such that the preimage f−1(B) of any Borel set B belongs to that σ-algebra, then f is said to be measurable. Measurable functions also form a vector space and an algebra as explained above.

Moreover, a set (family) of real-valued functions on X can actually define a σ-algebra on X 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 Ω are real-valued random variables.


Real numbers form a topological space and a complete metric space. Continuous real-valued functions (which implies that X 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, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.


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,[1] an open subset of them, or a smooth manifold.

Spaces of smooth functions also are vector spaces and algebras as explained above, and are a subclass 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.[2] Lp 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 Lp space, in the opposite direction for any f ∈ Lp(X) and xX which is not an atom, the value f(x) is undefined. Though, real-valued Lp spaces still have some of the structure explicated above. Each of Lp spaces is a vector space and have a partial order, and there exists a pointwise multiplication of "functions" which changes p, namely

\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.

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 and subharmonic functions (on Riemannian manifolds), analytic functions (usually of one or more real variables), algebraic functions (on real algebraic varieties), and polynomials (of one or more real variables).

See also


  1. ^ Different definitions of derivative exist in general, but for finite dimensions they result in equivalent definitions of classes of smooth functions.
  2. ^ Actually, a measure may have values in [0, +∞]: see extended real number line.

External links

Weisstein, Eric W., "Real Function", MathWorld.

This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.

Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.