Traditionally we speak of a function $f\left(x\right)$ if every $x$ satisfying ${x}_{1}<x<{x}_{2}$ is mapped to another number $y$ in a unique way;

Functionhere $y=f\left(x\right)$ is the mapping prescription, for example $y=sin\left(x\right)$ with real numbers of $y={z}^{n}$ with complex numbers $z,u$ and a real number $n$. For brevity one also writes $y\left(x\right)$ instead of $y=f\left(x\right)$.

In a more general manner one can define the concept of a function by mapping each element $a$ of a set $A$ uniquely to an element of the set $B$: the set $A$ is mapped to the set $B$ via the function f

$$B=f\left(A\right)$$

The $a\in A$ are also called preimage the $b\in B$ are called image or image points. Function and mapping are synonymous concepts.

The concept of mapping and function includes the uniqueness of the mapping.

The converse assumption is however not necessarily true: an image point $b$ can have many $preimages$ $a,{a}^{\prime}\cdots $. For the sine function there is for every $x$ a unique value $y=sin\left(x\right)$. Due to periodicity of the sine function modulo $2\pi $ every $y$ can be mapped to arbitrarily many $x$.

The sequences and series discussed in chapter 4 are examples for the mapping of discrete numbers – that means of functions, whose domain of definition for $x$ consists of discrete values $n$.

In general the domain of definition of the variable $a$ of a function will be continuous set, i.e. the set of real or complex numbers, or a limited region of one of these sets.

A function is continuous in the domain of definition of its preimage if the set of variables $a\in A$ is dense and in addition an arbitrarily small neighborhood of ${a}_{0}$ is mapped to a dense neighborhood of the image point ${b}_{0}$. Visually this means, that there are no gaps or jumps in the curve corresponding to $b\left(a\right)$.

The limit of the geometric series can be considered as the mapping of the continuous complex domain $a$ inside of the unit circle to the complex plane outside of the unit circle.

$$\begin{array}{c}\left|a\right|<1\hfill \\ z=f\left(a\right)=\underset{n\to \infty}{lim}\sum _{m=0}^{n}{a}^{m}=\frac{1}{1-a}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\to \phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}\phantom{\rule{0em}{0ex}}1\le z<\infty \hfill \\ \hfill \end{array}$$

The function is continuous in its domain of definition $\left|a\right|<1\phantom{\rule{0em}{0ex}}$.