### 5.1 Definition of Functions

Traditionally we speak of a function $Math content$ if every $Math content$ satisfying $Math content$ is mapped to another number $Math content$ in a unique way;

Functionhere $Math content$ is the mapping prescription, for example $Math content$ with real numbers of $Math content$ with complex numbers $Math content$ and a real number $Math content$. For brevity one also writes $Math content$ instead of $Math content$.

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

$Math content$

The $Math content$ are also called preimage the $Math content$ 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 $Math content$ can have many $Math content$ $Math content$. For the sine function there is for every $Math content$ a unique value $Math content$. Due to periodicity of the sine function modulo $Math content$ every $Math content$ can be mapped to arbitrarily many $Math content$.

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 $Math content$ consists of discrete values $Math content$.

In general the domain of definition of the variable $Math content$ 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 $Math content$ is dense and in addition an arbitrarily small neighborhood of $Math content$ is mapped to a dense neighborhood of the image point $Math content$. Visually this means, that there are no gaps or jumps in the curve corresponding to $Math content$.

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

$Math content$

The function is continuous in its domain of definition $Math content$.