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.

$f+g:x\mapsto f(x)+g(x)$ –

vector addition

$\mathbf {0} :x\mapsto 0$ –

additive identity

$cf:x\mapsto cf(x),\quad c\in \mathbb {R}$ –

scalar multiplication

$fg:x\mapsto f(x)g(x)$ – multiplication

pointwise

Let ${\mathcal {F}}(X,{\mathbb {R} })$ be the set of all functions from a set $X$ to real numbers $\mathbb {R}$ . Because $\mathbb {R}$ is a field, ${\mathcal {F}}(X,{\mathbb {R} })$ may be turned into a vector space and a commutative algebra over the reals with the following operations:

These operations extend to partial functions from $X$ to $\mathbb {R} ,$ with the restriction that the partial functions $f + g$ and $f g$ are defined only if the domains of $f$ and $g$ have a nonempty intersection; in this case, their domain is the intersection of the domains of $f$ and $g$ .

Also, since $\mathbb {R}$ is an ordered set, there is a partial order

on ${\mathcal {F}}(X,{\mathbb {R} }),$ which makes ${\mathcal {F}}(X,{\mathbb {R} })$ a partially ordered ring.

Continuous[edit]

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 in § Algebraic structure, and are a subclass of measurable functions because any topological space has the σ-algebra generated by open (or closed) sets.

Other appearances[edit]

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).

Real analysis

a major user of real-valued functions

Partial differential equations

Norm (mathematics)

Scalar (mathematics)

Apostol, Tom M. (1974). Mathematical Analysis (2nd ed.). Addison–Wesley. 978-0-201-00288-1.

ISBN

Real Analysis: Modern Techniques and Their Applications, Second Edition, John Wiley & Sons, Inc., 1999, ISBN 0-471-31716-0.

Gerald Folland

Rudin, Walter (1976). Principles of Mathematical Analysis (3rd ed.). New York: McGraw-Hill. 978-0-07-054235-8.

ISBN

External links[edit]

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