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

Hyperbolic cosine graph

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 ${\displaystyle D}$ ${\displaystyle F(D,\mathbb {R} )}$ ${\displaystyle \operatorname {Abb} (D,\mathbb {R} )}$ ${\displaystyle \mathbb {R} ^{D}}$ ${\displaystyle f}$ ${\displaystyle g}$

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

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 . ${\displaystyle \mathbb {R} ^{n}}$

literature

• Otto Forster: Analysis 1 . Differential and integral calculus of a variable. 11th, expanded edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-00316-6 , doi : 10.1007 / 978-3-658-00317-3 .
• Otto Forster: Analysis 2 . Differential calculus im , ordinary differential equations. 10th, improved edition. Springer Spectrum, Wiesbaden 2013, ISBN 978-3-658-02356-0 , doi : 10.1007 / 978-3-658-02357-7 . ${\displaystyle \mathbb {R} ^{n}}$
• Konrad Königsberger: Analysis 1 . 6th, revised edition. Springer-Verlag, Berlin Heidelberg New York 2004, ISBN 3-540-40371-X .

Web links

• Eric W. Weisstein : Real Function . In: MathWorld (English).
• LD Kudryavtsev: Real Function . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).