Numerical function

A numerical function is in mathematics a function whose function values extended real numbers , so real numbers included and are. ${\displaystyle -\infty }$ ${\displaystyle +\infty }$

If one considers a sequence of real functions, their supremum and infimum are generally not real. In measure theory , one therefore considers numerical functions .

definition

Let and denote the closure of the set of real numbers. One function ${\displaystyle \Omega \neq \emptyset }$ ${\displaystyle {\bar {\mathbb {R} }}=\mathbb {R} \cup \{-\infty ,\infty \}}$

${\displaystyle f\colon \Omega \to {\bar {\mathbb {R} }}}$

is called a numerical function .

Remarks

Every real-valued function is a numeric function, as are the extended functions . ${\displaystyle f\colon \Omega \to \mathbb {R} }$

Examples

• The constant function with , which can also be defined as or . ${\displaystyle f\colon \Omega \to {\bar {\mathbb {R} }}}$ ${\displaystyle f(\omega )\colon =c}$ ${\displaystyle c\in {\bar {\mathbb {R} }}}$ ${\displaystyle +\infty }$ ${\displaystyle -\infty }$
• The function

${\displaystyle f:\mathbb {R} \rightarrow {\bar {\mathbb {R} }},f(x)={\begin{cases}{\frac {1}{x^{2}}},&{\text{falls }}x\not =0\\+\infty ,&{\text{falls }}x=0\end{cases}}}$

is a numeric function. With the usual definition of convergence to ∞ , it is even continuous.

literature

• Klaus D. Schmidt: Measure and Probability . 2nd, revised edition. Springer, Heidelberg 2011, ISBN 978-3-642-21026-6 , doi : 10.1007 / 978-3-642-21026-6 .

Individual evidence

1. ↑ Klaus D. Schmidt: Measure and probability. ISBN 978-3-642-21026-6 , p. 91.