# Real-valued function

In mathematics, a real-valued function is a function whose function values ​​are real numbers . The term real function is closely related , but it is not clearly used in the literature. Real-valued functions can be found in almost all areas of mathematics, especially in analysis , functional analysis and optimization .

## definition

### Real-valued function

A real-valued function is a function

${\ displaystyle f \ colon D \ to \ mathbb {R}}$,

where the target set is the set of real numbers . The definition set is arbitrary. ${\ displaystyle D}$

### Real function

As with complex-valued and complex functions , the term real function is not used uniformly in the mathematical literature. Sometimes this term is synonymous with a real-valued function, sometimes only functions whose definition set is a subset of the real numbers, i.e. functions

${\ displaystyle f \ colon D \ to \ mathbb {R}}$,

where is. ${\ displaystyle D \ subseteq \ mathbb {R}}$

## Special cases

In the case of real-valued functions, there are generally no requirements placed on the structure of the definition set. If the definition set is to be restricted, a corresponding addition is appended to the term “real-valued function”. This is the name of a function, for example${\ displaystyle f \ colon D \ to \ mathbb {R}}$

• real-valued function of a real variable, if is,${\ displaystyle D \ subseteq \ mathbb {R}}$
• real-valued function of several real variables, if with is,${\ displaystyle D \ subseteq \ mathbb {R} ^ {n}}$${\ displaystyle n> 1}$
• real-valued function of a complex variable if is,${\ displaystyle D \ subseteq \ mathbb {C}}$
• real-valued function of several complex variables, if with is.${\ displaystyle D \ subseteq \ mathbb {C} ^ {n}}$${\ displaystyle n> 1}$

If is a subset of a real vector space , then a function is also called a (real-valued) functional . ${\ displaystyle D}$${\ displaystyle f \ colon D \ to \ mathbb {R}}$

## Examples

• The function is a real-valued function of a real variable.${\ displaystyle f (x) = x ^ {2}}$
• The function is a real-valued function of several real variables.${\ displaystyle f (x_ {1}, x_ {2}) = x_ {1} + x_ {2}}$
• The function that assigns its imaginary part to a complex number is a real-valued function of a complex variable.${\ displaystyle f (z) = \ operatorname {Im} (z)}$
• If the vector space of the symmetric real matrices , then the function is defined by a real-valued function.${\ displaystyle S ^ {n}}$${\ displaystyle f \ colon S ^ {n} \ to \ mathbb {R}}$${\ displaystyle f (A) = \ det (A)}$
• The null function is a real-valued function that is defined on any set. It assigns the number zero to each element.${\ displaystyle f (x) \ equiv 0}$

## Visualization

Graph of the functions (
paraboloid ) and ( absolute square )${\ displaystyle f (x, y) = x ^ {2} + y ^ {2}}$${\ displaystyle f (z) = | z | ^ {2}}$

The graph of a real-valued function of a real variable can be visualized by plotting the points in a two-dimensional coordinate system . To represent real-valued functions of two real variables, the points are entered in a three-dimensional coordinate system . With continuous functions, these representations form a curve or surface without jumps. With functions of two real variables, colors are sometimes used to visualize the function value. Real-valued functions of a complex variable can be represented in the same way as real-valued functions of two real variables. The imaginary part and the real part are taken as the first and second argument. ${\ displaystyle (x, f (x))}$${\ displaystyle (x_ {1}, x_ {2}, f (x_ {1}, x_ {2}))}$

## properties

### Algebraic properties

Addition of the sine function and the exponential function to with${\ displaystyle \ sin + \ exp: \ mathbb {R} \ to \ mathbb {R}}$${\ displaystyle (\ sin + \ exp) (x) = \ sin (x) + \ exp (x)}$

The set of all real-valued functions over a given set forms a real vector space , which is denoted by , or . The sum of two real-valued functions and is defined by ${\ displaystyle D}$${\ displaystyle F (D, \ mathbb {R})}$${\ displaystyle \ operatorname {Fig} (D, \ mathbb {R})}$${\ displaystyle \ mathbb {R} ^ {D}}$${\ displaystyle f}$${\ displaystyle g}$

${\ displaystyle (f + g) (x) = f (x) + g (x)}$

for all and the product of a real-valued function with a real number by ${\ displaystyle x \ in D}$${\ displaystyle f}$${\ displaystyle c \ in \ mathbb {R}}$

${\ displaystyle (c \ cdot f) (x) = c \ cdot f (x)}$

for everyone . These vector spaces are called real function spaces . They play an important role in linear algebra and calculus . With addition and point-wise multiplication defined by ${\ displaystyle x \ in D}$

${\ displaystyle (f \ cdot g) (x) = f (x) \ cdot g (x)}$

for all the real-valued functions over the set form a commutative ring . With all three links, the real-valued functions form a real algebra . ${\ displaystyle x \ in D}$${\ displaystyle D}$

### Analytical properties

A real-valued function is called bounded if there is a bound such that ${\ displaystyle f \ colon D \ to \ mathbb {R}}$${\ displaystyle M}$

${\ displaystyle | f (x) | \ leq M}$

is for everyone . The set of bounded real-valued functions forms with the supremum norm${\ displaystyle x \ in D}$${\ displaystyle B (D, \ mathbb {R})}$

${\ displaystyle \ | f \ | _ {\ infty}: = \ sup _ {x \ in D} | f (x) |}$

a standardized space . Since the real numbers are complete , it is even a Banach space . A sequence of real-valued functions with for is called uniformly bounded if every term in the sequence is a bounded function and the sequence ${\ displaystyle (f_ {1}, f_ {2}, \ ldots)}$${\ displaystyle f_ {n} \ colon D \ to \ mathbb {R}}$${\ displaystyle n = 1,2, \ ldots}$

${\ displaystyle (\ | f_ {1} \ | _ {\ infty}, \ | f_ {2} \ | _ {\ infty}, \ ldots)}$

is a bounded sequence of real numbers. A sequence of real-valued functions is called pointwise bounded if the real number sequence for all${\ displaystyle x \ in D}$

${\ displaystyle (f_ {1} (x), f_ {2} (x), \ ldots)}$

is limited. A uniformly bounded sequence of real-valued functions is always bounded pointwise, but the converse does not have to hold. A sequence of real-valued functions is called uniformly convergent to a real-valued function if ${\ displaystyle f \ colon D \ to \ mathbb {R}}$

${\ displaystyle \ lim _ {n \ to \ infty} \ | f_ {n} -f \ | _ {\ infty} = 0}$

applies. Correspondingly, a sequence of real-valued functions is called pointwise convergent to a real-valued function if for all${\ displaystyle f \ colon D \ to \ mathbb {R}}$${\ displaystyle x \ in D}$

${\ displaystyle \ lim _ {n \ to \ infty} (f_ {n} (x) -f (x)) = 0}$

applies. Here, too, the uniform convergence results in point-wise convergence, but not the reverse. Further analytical properties, such as continuity , differentiability or integratability , require at least a topological , metric or mass-theoretical structure on the definition set .

### Order properties

After the real numbers are ordered , the partial order can be used for real-valued functions

${\ displaystyle f \ leq g \ Leftrightarrow \ forall x \ in D: f (x) \ leq g (x)}$

define. A sequence of real-valued functions with is then called monotonically increasing . The partial order is analogous ${\ displaystyle (f_ {1}, f_ {2}, \ ldots)}$${\ displaystyle f_ {1} \ leq f_ {2} \ leq \ ldots}$

${\ displaystyle f \ geq g \ Leftrightarrow \ forall x \ in D: f (x) \ geq g (x)}$

defined and a sequence of real-valued functions with is then monotonically decreasing . ${\ displaystyle f_ {1} \ geq f_ {2} \ geq \ ldots}$

## Generalizations

The real-vector- valued functions form a generalization of the real-valued functions . These are functions that are mapped in the. The vector-valued functions that map into arbitrary vector spaces are even more general . Functions that take complex function values ​​are called complex-valued functions . ${\ displaystyle \ mathbb {R} ^ {n}}$